An Algebraic Approach to Fourier Transformation
AAN ALGEBRAIC APPROACH TOFOURIER TRANSFORMATION
MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ Abstract.
The notion of Fourier transformation is described froman algebraic perspective that lends itself to applications in Sym-bolic Computation. We build the algebraic structures on the basisof a given Heisenberg group (in the general sense of nilquadraticgroups enjoying a splitting property); this includes in particular thewhole gamut of Pontryagin duality. The free objects in the cor-responding categories are determined, and various examples aregiven. As a first step towards Symbolic Computation, we studytwo constructive examples in some detail—the Gaussians (with andwithout polynomial factors) and the hyperbolic secant algebra.
Date : September 28, 2020. ∗ Research funded by the Austrian Science Fund (FWF) under Grant P30052. a r X i v : . [ m a t h . R A ] S e p MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ Contents
1. Introduction 21.1. Motivation from Algorithmic Analysis. 21.2. Overview of the paper. 31.3. Terminological conventions and notation. 52. Heisenberg Modules and Algebras 72.1. Review of Heisenberg Groups. 72.2. The Category of Heisenberg Algebras. 202.3. The Heisenberg Twist. 262.4. Free Heisenberg Modules and Algebras. 323. Fourier Operators in Algebra 393.1. The Notion of Fourier Doublets. 393.2. Classical Pontryagin Duality. 443.3. Fourier Inversion. 563.4. The Schwartz Class for Pontryagin Duality. 693.5. Classical Schwartz Functions and the Weyl Algebra. 764. Constructive Fourier Analysis via Schwartz Functions 844.1. A Minimal Subalgebra of the Schwartz Class. 844.2. The Gelfond Field for Coefficients. 984.3. The Rational Fourier Singlet of Gaussians. 1044.4. Holonomic Fourier Extensions. 1084.5. The Hyperbolic Fourier Doublet. 118Acknowledgments 121References 1211.
Introduction
Motivation from Algorithmic Analysis.
In this paper we havedeveloped an algebraic theory of Fourier transforms, which is intended(but not restricted) to serve as a convenient framework for SymbolicComputation. The goal we have in mind is to build up a symbolicoperator calculus to determine the Green’s operators of certain bound-ary problems for linear partial differential equations (LPDE). In thisapplication, the role of the Fourier operator is roughly analogous tothe indefinite integration operator, denoted by (cid:114) or A in our earlierpapers [93, 96, 97, 98, 28]. While it is possible to extend the opera-tor calculus of these papers to the multivariate case by encoding thesubstitution rule of integration into a suitable relation ideal [94], suchoperators are usually not sufficiently expressive for capturing Green’soperators for LPDE boundary problems. N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 3
In fact, even for constant coefficients, only the degenerate case ofcompletely reducible characteristic polynomials [95] is amenable to a di-rect treatment via (cid:114) x , (cid:114) y , . . . in conjunction with substitutions and mul-tiplication operators. In contrast, Fourier transforms are well known toprovide feasible tools for expressing the solution operators in the caseof LPDE with constant coefficients , so they form a reasonable basisfor developing algorithms to compute Green’s operators in an algebraicoperator calculus. Of course, the full specification of such an algo-rithm will also fix a suitabe class of boundary conditions with certainconstraints to ensure well-posedness.Apart from the principal goal of setting up an operator calculus, wewould like to mention two other potential application areas: The firstwould be a kind of algebraic structure theory for Fourier transforms,somewhat akin to differential Galois theory [111]. Some first steps inthis direction may be perceived in the material of Section 4.The other major application area in Symbolic Computation wouldbe to build up an algorithm for Fourier transforms , perhaps remotelyreminiscent of the Risch algorithm [88]. Guided by the structure theoryaddressed earlier, such an algorithm would either express the Fouriertransform of a given function (as an element of the “signal space” ac-cording to the terminology in the present paper) in terms of a fixedtarget domain (here called “spectral space”), or otherwise report thatthis is not possible. In the latter case, adjunction of new elementswould lead to larger spaces/algebras that contain the desired functions.Clearly, it will take considerable effort to build up such a theory, butwe believe that the structures developed in this paper could provide auseful basis for such an undertaking.1.2.
Overview of the paper.
The material of this paper is to someextent coupled with that of its companion paper [61], where a moregeneral view of Heisenberg groups is developed, mainly from the per-spective of homological algebra. In the present paper, we focus onthe structures actually arising in constructive analysis as indicated inthe previous subsection. Whenever we refer to specific places in thecompantion paper, we will use the shorthand (cid:74) . . . (cid:75) for [61, . . . ].The remainder of this paper is divided into three sections. Just asfield theory is built up before exploring differential fields (by introduc-ing the notion of derivation), we first build up the theory of Heisenbergmodules , based on a fixed Heisenberg group, in §2. From there we de-velop the theory of
Fourier doublets (by introducing an algebraic notionof Fourier operator in §3); in this way we can capture the various func-tion spaces—with or without multiplication—on which various species
MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ of Fourier operators act. Finally, we investigate in §4 some particularinstances of Fourier doublets from the viewpoint of Symbolic Compu-tation , giving special attention to the important example of Gaussianfunctions (with or without polynomial factors adjoined).Let us now go through these three sections in some more detail.We start in §2.1 by introducing the particular notion of Heisenberggroup that is explored in this paper, with special emphasis on the cru-cial examples coming from Pontryagin duality and the so-called sym-plectic Heisenberg groups. Fixing some specific Heisenberg group H ,the category of Heisenberg modules over H along with the more en-riched categories of Heisenberg algebras (featuring one multiplication)and
Heisenberg twain algebras (featuring two multiplications) are set upin §2.2. Some basic categorical properties are revealed (monoidal struc-ture, coproduct). In §2.3, we turn to a feature of Heisenberg groupsthat is more directly relevant to the goal of building algebraic Fourierstructures—the
Heisenberg twist J , an involutive antihomomorphismon a Heisenberg group. Roughly speaking (confer Remark 29), Fourieroperators might be likened to conjugate-linear maps between complexvector spaces (where the involution J is complex conjugation). Return-ing to categorical considerations in §2.4, we explore next the free objects in various categories of Heisenberg modules and (twain) algebras.In §3.1, we define a Fourier doublet as a pair consisting of two Heisen-berg modules or two Heisenberg (twain) algebras—referred to as sig-nal/spectral spaces, respectively—harnessed together with a Fourieroperator between them. The role of the Heisenberg twist is elucidatedby way of the
Heisenberg clock (Figure 2). We set up suitable categoriesand their free objects for the various sorts of Fourier doublets. Theall-important case of
Pontryagin duality is explored in §3.2, includingvarious well-known Fourier operators as special cases (Fourier integral,Fourier series, discrete-time Fourier transform, discrete Fourier trans-form). While this includes multivariate functions, one may alterna-tively reduce them to univariate functions via the tensor product. Theclosely related symplectic Fourier transform is also mentioned. Thetopic of
Fourier inversion is initiated in §3.3, along with crucial ex-amples such as an “invertible subdoublet” of the canonical L setting,an extension to L functions, and a Fourier doublet with measuresas signals. For the above-mentioned special cases, Fourier inversionexhibits the familiar sign change in the “exponential” (= character).Arguably the most convenient setting for Fourier analysis, Schwartzfunctions —along with the corresponding tempered distributions—areintroduced in §3.4 for the general case of Pontryagin duality (wherethey are more properly named Schwartz-Bruhat). We sketch the setup
N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 5 of a ring of differential operators following [6]. This generalizes the ac-tion of the Weyl algebra on the classical Schwartz space over R n , whichwe investigate in §3.5 from the algebraic perspective. This includes inparticular the crucial differentiation law underlying applications for dif-ferential equations (“differentiation is multiplication by the symbol”).The relation beteen Fourier and Laplace transformation—for generalPontryagin duality—is briefly mentioned.We start the consideration of Symbolic Computation aspects in §4.1by extracting from Schwartz space all the Gaussian functions . Whilethis is not yet fully constructive, it allows to come up with explicitformulae for the twain algebra operations and the Fourier operator.Linear independence of the Gaussians and related functions in ensuredby employing a general asymptotic scale (we develop this approach insome detail since it may come in handy later when considering moregeneral cases). The resulting Heisenberg algebra is viewed as a semi-group extension similar to the group extensions studied in group coho-mology (again this may be of value for extensions to be contemplatedat a later stage). For making the Gaussian doublet fully algorithmic,we set up an effective number field in §4.2. We call it the
Gelfondfield since it hinges crucially on Gelfond’s famous transcendency resultfor e π . Restricting to this field in §4.3, we trim down the Gaussiandoublet to a fully algorithmic domain that is shown to be generatedwithin Schwartz space by a single Gaussian (essentially the probabilitydensity of the normal distribution). It is then shown to be a quotient ofa free twain algebra modulo certain identities relating convolution andpointwise multiplication of Gaussians. For going beyond the Gaussiandoublet in Symbolic Computation, we look at the standard approachvia holonomic functions (and holonomic distributions) in §4.4. Thishas the decisive advantage that the Weyl action mentioned above isalso available. Some specific examples are given, in particular the im-portant case of adjoining the Gaussian doublet by polynomial factors.The Fourier integral is given in terms of Hermite functions , which areanalyzed by umbral algebra. As another example, in §4.5 we set upa Fourier doublet generated by the hyperbolic secent (which is its ownFourier transform). They are viewed as subdoublets of Schwartz space,and explicit formulae are available to some extent. More work will beneeded at this point to obtain a more explicit description.1.3.
Terminological conventions and notation.
The category of abelian groups is denoted by Ab , the category of left and right modules over a ring R by R Mod and
Mod R , respectively. Algebras generallyassumed commutative but nonunital; the corresponding categories are
MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ written R Alg and
Alg R . By an LCA group we mean a locally compactabelian group (where locally compact includes Hausdorff). An invo-lution is an automorphism—of groups or algebras—that is inverse toitself. The nonzero elements of R are denoted by R × , its group of units by R ∗ . The path algebra for a quiver Q over R is denoted by R (cid:76) Q (cid:77) .If Q is not endowed with a quiver structure, it is taken as a discretequiver (no arrows). This is to be contrasted to the group algebra R [ G ] over the ring R , for any given group G .A nilpotent group of nilpotency class at most two will be called nilquadratic . The center of a group G is denoted by (cid:90) G . If Γ and G are any abelian groups, we write their tensor product as Γ ⊗ G := Γ ⊗ Z G .A bilinear form on these groups is a Z -bilinear map β : Γ × G → T ,viewing abelian groups as Z -modules (hence a Z -linear map Γ ⊗ G → T ). We will often write the action of such a form as (cid:104) ξ | x (cid:105) β = β ( ξ, x ) ,suppressing the subscript β when it is obvious from the context. Wecall β or a duality if β ( ξ, − ) and β ( − , x ) are injective for all ξ ∈ Γ and x ∈ G . The term bicharacter is usually taken as synonymous withthe notion of bilinear form (with the exception of Example 7).Given a R -module M , an n -form is a bilinear map M n → R ; for n = 2 one either suppresses n or speaks of a bilinear form . An alternatingform is a bilinear map ω : M ⊕ M → R such that ω ( x, x ) = 0 forall x ∈ M ; it is called a symplectic form if it is moreoever, againin the sense that ω ( x, − ) : M → M is injective for all x ∈ M . Onerefers to the structure ( M, ω ) as an alternating or symplectic module,respectively. The symmetric algebra over M is denoted by Sym( M ) ∼ = T ( M ) /I ( M ) , where I ( M ) (cid:69) T ( M ) is the ideal generated by { x ⊗ y − y ⊗ x | x, y ∈ M } . In case of ambiguity, the base ring R maybe explicated in writing Sym R ( M ) ∼ = T R ( M ) /I R ( M ) . Note that for aset X one has R [ X ] = Sym R ( RX ) , where RX is the free R -moduleover the basis X . Given a homomorphism σ : R (cid:48) → R , an R -module M becomes an R (cid:48) -module M [ σ ] via scalar restriction along σ ; one alsocalls M [ σ ] a twisted module .If K is a field, we write the vector space of column vectors as K n , thatof row vectors as K n . The n × n identity matrix is denoted by I n . Weshall sometimes employ the abbreviation x − := x − for the reciprocal (when it exists) of an element x in a multiplicatively written monoid.The unit interval is denoted by I = [0 , . Given numbers δ , δ ∈ R ,we use x ± δ y as an abbreviation for their weighted sum or difference ( δ x ± δ y ) / ( δ + δ ) ; note that + δ is in general not commutative.We introduce also the harmonic sum (cid:1) on R > as x (cid:1) y := xy/ ( x + y ) to create the harmonic semigroup ( R > , (cid:1) ) ∼ = ( R > , +) , with the iso-morphism given by i : R > → R > , x (cid:55)→ x − . N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 7 Heisenberg Modules and Algebras
Review of Heisenberg Groups.
As outlined in the Introduc-tion, our treatment of Fourier operators is based on Heisenberg modulesand Heisenberg algebras. These in turn are built on the central notionof
Heisenberg group . In the literature [20] [68] [85] [84] [113] one findsseveral versions of this idea, exhibiting various degrees of generality. Inthe present paper, we shall use a definition close to [20, Def. 3.1], sincethis definition remains entirely within algebraic confines, admitting alevel of generality sufficient for our current goal: to forge algebraic andalgorithmic tools that support calculations with Fourier operators. InDefinition (cid:74) ?? (cid:75) , we provide a somewhat wider notion of Heisenberggroup, with a view towards homological question.For preparing our definition, recall that a Heisenberg group—evenunder more general conceptions—is always a nilquadratic extension,meaning a central extension H of an abelian group P by anotherabelian group T . Whereas the phase space P is written additively,the torus T is conventionally presented in multiplicative notation (inview of its most prominent specimen T = S ⊂ C × ). The correspond-ing short exact sequence E : T ι (cid:26) H π (cid:16) P gives rise to a symplecticstructure by way of the commutator form ω E ; we refer the reader to§ (cid:74) ?? (cid:75) for some basic terminology in this area, in particular the notionsof (co)symplectic subgroups and Lagrangian subgroups.At this point we shall only review the notions immediately enteringthe definition of Heisenberg group given below. Recall that maximalabelian subgroups ˜ G ≤ H are in bijective correspondence with La-grangian subgroups G ≤ P via the projection π : H → P . A pair ( ˜ G, ˜Γ) of maximal abelian subgroups is called an abelian bisection of H over T if ˜ G ˜Γ = H and ˜ G ∩ ˜Γ = ˆ T := ι ( T ) ; it corresponds under π to a La-grangian bisection of H , meaning a direct decomposition of P intoLagrangian subgroups. It is clear that π then descends to isomor-phisms H/ ˜ G ∼ = Γ and H/ ˜Γ ∼ = G .Note that ˜ G and ˜Γ both contain the embedded torus ˆ T := ι ( T ) . If thelatter splits off as a direct summand in ˜ G as well as ˜Γ , we call ( ˜ G, ˜Γ) an abelian splitting and ( G, Γ) a Lagrangian splitting . As a consequence, ˆ T has direct complements in ˜ G and ˜Γ , which are easily seen to beisomorphic to G and Γ , respectively. Hence we shall take the libertyof designating those direct complements by G, Γ ≤ H as well. Wecan now introduce Heisenberg groups in the form they are used in thispaper. MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ Definition 1. A Heisenberg group ( H ; ˜ G, ˜Γ) is a nilquadratic group H with an abelian splitting ( ˜ G, ˜Γ) .As usual in such cases, the abelian splitting ( ˜ G, ˜Γ) is sometimessuppressed when speaking of “the Heisenberg group H ”, but ( ˜ G, ˜Γ) must nevertheless be viewed as part of the data: As we shall see below(Example 17), existence and uniqueness of abelian splittings may fail for a general nilquadratic group. In contrast, we shall soon learn thatthe (implicit) choice of complements G, Γ ≤ H of ˆ T has no influenceon the Heisenberg group.The Heisenberg groups form a category Hei with objects short exactsequences H : T ι (cid:26) H π (cid:16) G ⊕ Γ . If H (cid:48) : T (cid:48) ι (cid:48) (cid:26) H (cid:48) π (cid:48) (cid:16) G (cid:48) ⊕ Γ (cid:48) is an-other Heisenberg object, a Heisenberg morphism ( t, h, g × γ ) : H → H (cid:48) consists of group homomorphisms t : T → T (cid:48) and h : H → H (cid:48) as wellas g ⊕ γ : G ⊕ Γ → G (cid:48) ⊕ Γ (cid:48) such that the diagram(1) (cid:47) (cid:47) T (cid:47) (cid:47) t (cid:15) (cid:15) H (cid:47) (cid:47) h (cid:15) (cid:15) G ⊕ Γ (cid:47) (cid:47) g ⊕ γ (cid:15) (cid:15) (cid:47) (cid:47) T (cid:48) (cid:47) (cid:47) H (cid:48) (cid:47) (cid:47) G (cid:48) ⊕ Γ (cid:48) (cid:47) (cid:47) commutes. Thus a Heisenberg morphism is just a morphism of shortexact sequences that respects the Lagrangian splittings.The Heisenberg groups (1) with fixed Lagrangian splitting ( G, Γ) form a subclass within all nilquadratic extensions, corresponding to acertain subgroup H G, Γ ( P, T ) ≤ H ( P, T ) under cohomology. As for allnilquadratic extensions, the Universal Coefficient Theorem induces ashort extact sequence(2) (cid:47) (cid:47) Ext ( P, T ) j (cid:47) (cid:47) H ( P, T ) q (cid:47) (cid:47) Ω ( P, T ) (cid:47) (cid:47) which splits, albeit not naturally. Here j is the natural embedding ofabelian (= symmetric) cocycles while q is the skewing map [ γ ] (cid:55)→ ω with ω ( z, w ) = γ ( z, w ) /γ ( w, z ) . In the (rare) event that T is uniquely -divisible [116, Thm. 5.4] in the sense that for every c ∈ T thereexists a unique d ∈ T with d = c , one may choose a canonical sec-tion Ω ( P, T ) → H ( P, T ) of q as follows. Writing √ for the inverse ofthe squaring homomorphism, one sends ω ∈ Ω ( P, T ) to γ ∈ H ( P, T ) defined by γ ( z, w ) = (cid:112) ω ( z, w ) . In other words, one can recover theequivalence class of the group extension H of (1) from its commutatorform ω via H ∼ = [ H ] ω := T × √ ω P ; see Lemma 3 below for the case ofHeisenberg groups.There is an alternative perspective on Heisenberg groups that empha-sizes their constructive nature. Given any bilinear form β : Γ × G → T N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 9 from abelian groups G and Γ to another abelian group T , one mayconstruct the phase action (3) (cid:67) : Γ → Aut(
T G ) , (cid:67) ξ cx = c (cid:104) x | ξ (cid:105) x, where T G denotes the direct product of T with G , whose elements wechoose to write as cx ∈ T G with c ∈ T and x ∈ G . It is clear that thephase action (3) is faithful iff (cid:104)|(cid:105) β is a duality. Definition 2.
Let β : Γ × G → T be a bilinear form. Then the Heisen-berg group assciated to β is the semidirect product H ( β ) := T G (cid:111) Γ with respect to the phase action (3).As has been shown in [20, Thm. 3.4] for a slightly restricted set-ting (with torus T = (cid:90) ( H ) and with β being a duality) and in Def-inition (cid:74) ?? (cid:75) for somewhat more general cases (requiring only bisec-tions rather than splittings), the Heisenberg groups are exactly thoseof the form H ( β ) . More precisely, for each Heisenberg group ( H ; ˜ G, ˜Γ) there is a unique bilinear form β : Γ × G → T such that H ∼ = H ( β ) in Hei , where H ( β ) is endowed with the Lagrangian splitting ( ˜ G, ˜Γ) with ˜ G := T × G and ˜Γ := T × Γ . This also reiterates our observationabove that the abelian splitting is part of the data for a Heisenberggroup as it corresponds to the choice of bilinear form. On the otherhand, the choice of complements of ˆ T in both ˜ G and ˜Γ is immaterial; itmerely determines the particular isomorphism between H and H ( β ) .In this paper, we represent Heisenberg groups in the form H ( β ) , witha bilinear form β : Γ × G → T . Applied to arguments ( ξ, x ) ∈ Γ × G ,the former is written as (cid:104) ξ | x (cid:105) β , with the index β omitted when thecontext makes it clear. Motivated by the fundamental example of sym-plectic duality (see Remark 16 below), the elements of G will be called positions , those of Γ as momenta . If β is nondegenerate, we can iden-tify positions x ∈ G with their position characters (cid:104)−| x (cid:105) : Γ → T givenby ξ (cid:55)→ (cid:104) ξ | x (cid:105) , and momenta ξ ∈ Γ with their momentum characters (cid:104) ξ |−(cid:105) : G → T , x (cid:55)→ (cid:104) ξ | x (cid:105) .The isomorphism H ∼ = [ H ] ω := T × √ ω P mentioned above takeson the following form in case of a Heisenberg group H = H ( β ) . Notethat [ H ] ω may be viewed as a canonical representation of all Heisenbergextensions (1) with fixed commutator form ω : P × P → T , but [ H ] ω is not itself a Heisenberg group in our sense of Definition 1. Due to itsmost important instances (see Example 15 below), we shall call it the symplectic Heisenberg group . Lemma 3. If β : Γ × G → T is a bilinear form over a uniquely -divisible torus T with square root √ , the Heisenberg extension (1) is ∗ equivalent to the extension T (cid:26) [ H ] ω (cid:16) P via the polarization map P G, Γ : [ H ] ω ∼ −→ H ( β ) given by ( c ; x, ξ ) (cid:55)→ c (cid:112) (cid:104) ξ | x (cid:105) ( x, ξ ) .Proof. Clearly, Φ is inverted by c ( x, ξ ) (cid:55)→ ( c (cid:112) (cid:104) ξ | x (cid:105) − ; x, ξ ) , and it isa group homomorphism as one checks immediately. Hence (1 T , Φ , P ) is an equivalence from the extension T (cid:26) T × √ ω P (cid:16) P to theHeisenberg extension (1). (cid:3) As noted above, there is no canonical section of the entire skewingmap q : H ( P, T ) → Ω ( P, T ) if the torus T is not uniquely -divisible.In the presence of a Lagrangian splitting P = G ⊕ Γ , however, one canstill recover the Heisenberg cocycles γ ∈ H ( P, T ) from their commu-tator forms ω ∈ Ω ( P, T ) . In fact, the bilinear cocycles γ ∈ Z ( P, T ) with γ ( z, w ) = 0 for z ∈ G or w ∈ Γ form a subgroup Z G, Γ ( P, T ) holding canonical representatives for H G, Γ ; see Proposition (cid:74) ?? (cid:75) . Sim-ilarly, one may introduce the subgroup Ω G, Γ ( P, T ) of skew-symmetricforms ω ∈ Ω ( P, T ) that vanish on Γ × Γ and G × G . Let Γ ι Γ (cid:47) (cid:47) P π Γ (cid:111) (cid:111) π G (cid:47) (cid:47) G ι G (cid:111) (cid:111) be the corresponding direct sum diagram with associated projectors P Γ = ι Γ π Γ , P G = ι G π G : P → P . Then we have the following commu-tative diagram of group isomorphisms: Ω G, Γ ( P, T ) ( p Γ × p G ) ∗ (cid:5) (cid:5) ( ι Γ × ι G ) ∗ (cid:24) (cid:24) Z G, Γ ( P, T ) ( ι Γ × ι G ) ∗ (cid:47) (cid:47) q (cid:51) (cid:51) (Γ ⊗ G ) ∗ ˜ q (cid:107) (cid:107) ( π Γ × π G ) ∗ (cid:111) (cid:111) Here ˜ q maps a bilinear map β : Γ × G → T to the skew-symmetricform ω = ( π Γ × π G ) ∗ ( β ) − ( π G × π Γ ) ∗ ( β (cid:48) ) . In other words, one appliesantisymmetrization in the form ω ( x + ξ, y + η ) = β ( ξ, y ) − β ( η, x ) .According to the above diagram, Heisenberg cocycles γ ∈ H G, Γ ( P, T ) ,their commutator forms ω ∈ Ω G, Γ ( P, T ) and the associated dualities β : Γ × G → T all contain the same information in view of a Lagrangiansplitting P = G ⊕ Γ .Before considering some examples of Heisenberg groups, let us firsthave a closer look at the group operations. According to Definition 2,the multiplication in a Heisenberg group is given by(4) c ( x, ξ ) · c (cid:48) ( x (cid:48) , ξ (cid:48) ) = cc (cid:48) (cid:104) ξ | x (cid:48) (cid:105) ( x + x (cid:48) , ξ + ξ (cid:48) ) , N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 11 the unit element by , and the inversion map by(5) (cid:0) c ( x, ξ ) (cid:1) − = c − (cid:104) ξ | x (cid:105) ( − x, − ξ ) for any c ( x, ξ ) ∈ H ( β ) and c (cid:48) ( x (cid:48) , ξ (cid:48) ) ∈ H ( β ) . The group law (4)may also be expressed as a kind of schematic matrix multiplication ifone identifies the triples with × matrices such that their productscorrespond via(6) c ( x, ξ ) · c (cid:48) ( x (cid:48) , ξ (cid:48) ) ↔ ξ c x · ξ (cid:48) c (cid:48) x (cid:48) . Here one should keep in mind that the addition of the upper rightmatrix elements corresponds to the (multiplicatively written) groupoperation in T . Furthermore, one agrees that a left matrix element ξ ∈ Γ multiplies with a right matrix element x (cid:48) ∈ G to yield (cid:104) ξ | x (cid:48) (cid:105) ∈ T ; allother products are trivial since they involve or .For closer analysis we introduce the bilinear category as the commacategory Bi := ⊗ ↓ Ab where ⊗ : Ab × Ab → Ab is the tensorproduct. By the characteristic property of the latter, we may view theobjects Γ ⊗ G β → T of Bi as bilinear forms (also known as bicharacters ).For brevity, we shall also write these as (cid:104) Γ | G (cid:105) β , omitting the subscript β when it is clear from the context. A morphism ( γ × g, t ) : β → β (cid:48) between β : Γ × G → T and β (cid:48) : Γ (cid:48) × G (cid:48) → T (cid:48) is given by homomorphisms γ × g : Γ × G → Γ (cid:48) × G (cid:48) and t : T → T (cid:48) such that t (cid:104) ξ | x (cid:105) β = (cid:104) γξ | gx (cid:105) β (cid:48) for all ( ξ, x ) ∈ Γ × G . One may check that Bi is a bicomplete abeliancategory. Fixing a torus T , we write Bi ( T ) for the subcategory of Bi consisting of bilinear forms Γ ⊗ G → T with arbitrary Γ , G ∈ Ab ,having morphisms ( γ × g, T ) in between them. We shall subsequentlysuppress the identity T , referring to γ × g as a morphism in Bi ( T ) .There is a tensor product on Bi ( T ) that sends a pair β : Γ × G → T and β (cid:48) : Γ (cid:48) × G (cid:48) → T of bilinear forms to the bilinear form β ⊗ β (cid:48) : (Γ ⊕ Γ (cid:48) ) × ( G ⊕ G (cid:48) ) → T with (cid:104) ξ, ξ (cid:48) | x, x (cid:48) (cid:105) β ⊗ β (cid:48) := (cid:104) ξ | x (cid:105) β (cid:104) ξ (cid:48) | x (cid:48) (cid:105) β (cid:48) and a pair γ × g and γ (cid:48) × g (cid:48) of morphisms in Bi ( T ) to ( γ ⊕ γ (cid:48) ) × ( g ⊕ g (cid:48) ) . As one sees immediately—associators and unitors just shufflingparentheses—this gives a symmetric monoidal structure on Bi ( T ) withunit object the trivial bilinear form ⊕ → T .We write Du for the full subcategory of Bi consisting of dualities (i.e. nondegenerate bilinear forms). It is easy to see that Du ( T ) is asymmetric monoidal subcategory of Bi . ∗ Remark 4.
It is sometimes useful to think of Du as fibered over Ab by sending a duality β : Γ × G → T to its torus T . The fiber Du ( T ) has finite coproducts (suppressing some cartesian products for clarity):If β (cid:48) : Γ (cid:48) × G (cid:48) → T is another object of Du ( T ) , their coproduct β × β (cid:48) : ΓΓ (cid:48) × GG (cid:48) → T is the duality ( ξξ (cid:48) , xx (cid:48) ) (cid:55)→ β ( ξ, x ) β (cid:48) ( ξ (cid:48) , x (cid:48) ) withinjections ι × ι : Γ G → ΓΓ (cid:48) × GG (cid:48) and ι × ι : Γ (cid:48) G (cid:48) → ΓΓ (cid:48) × GG (cid:48) . (cid:125) As detailed in Proposition (cid:74) ?? (cid:75) , the association β (cid:55)→ H ( β ) is a func-tor H : Du → Hei , which is adjoint to the functor B : Hei → Du extracting the restriction β : Γ × G → T of the commutator form ω E of a given Heisenberg extension. In fact, more is true[71, p. 94]: Thefunctor B is left-adjoint left-inverse to the functor H , so the latter isan isomorphism of Du to the reflective subcategory { H ( β ) | β ∈ Du } of Hei , taking license to use curly braces for class formation. In otherwords, we have an adjoint equivalence whose counit is trivial and whoseunit may be viewed as a canonical simplifier [27]: Every Heisenberggroup H is isomorphic to exactly one H ( β ) ∼ = H that acts as the canon-ical representative of its equivalence class. Thinking of B : Hei → Du as a very special fibration (each fiber consisting of equivalent objects),the unit creates a cross-section through the fibers.This is why in the following we usually write Heisenberg groups incanonical form H ( β ) = T G (cid:111) Γ . The full subcategory of Hei consistingof Heisenberg groups over β ∈ Du is denoted by Hei β . Moreover, weshall identify G and Γ as subgroups of H ( β ) . Similarly, the torus T isidentified with its embedding ˆ T in H ( β ) . Example 5.
The most important examples of Heisenberg groups arebased on
Pontryagin duality . Given a locally compact abelian group G ,its characters in the classical sense are the continuous homomorphismsfrom G into the complex torus T ⊆ C . The collection ˆ G of all char-acters is then again an abelian group known as the dual group of G ,and the famous Pontryagin duality theorem implies that the naturalpairing (cid:36) : ˆ G × G → T with (cid:104) ξ | x (cid:105) (cid:36) := ξ ( x ) is a nondegenerate bilinearform; confer for example Theorem 1.7.2 of [100] or [75, §4]. In thisway, the Heisenberg group H ( (cid:36) ) may be associated to every locallycompact group G , as in André Weil’s definition (4) of [119, p. 149],where the Heisenberg group H ( (cid:36) ) is denoted by A ( G ) .Note that this also includes plain groups G without topology sinceone may always endow them with the discrete topology. In this case,the character group ˆ G is the group of all homomorphisms G → T ,no matter what the topology on T may be. Thus one may view thealgebraic setting as contained in a topological frame (where ˆ G is anLCA group). Note that a bilinear form β : Γ × G → T is nondegenerate N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 13 iff one has embeddings
G (cid:44) → ˆΓ and Γ (cid:44) → ˆ G for encoding positions andmomenta by their characters. // Example 6.
One of the most famous examples of Pontryagin duality isgiven by the map T × Z → T defined by (cid:104) ξ | x (cid:105) := ξ x . It is clear that thismay be extended to T n × Z n → T with (cid:104) ξ | x (cid:105) = ξ x = ξ x · · · ξ x n n . Werefer to this example as the torus duality Tor = (cid:104) T n | Z n (cid:105) . Interchangingpositions and momenta, we obtain the conjugate torus duality Tor (cid:48) = (cid:104) Z n | T n (cid:105) . Their basic difference will be seen in Example 40c, their closerelationship at the end of Example 53. // Another basic example of Pontryagin duality is finite-dimensional normed vector spaces (Euclidean vector spaces with norm topology).In this case, all linear functionals are automatically continuous, so onemay in fact ignore the topology altogether and consider the generalcase of vector spaces with bilinear forms.
Example 7.
More precisely, we study now bilinear forms Γ × G → T where G = Γ is an n -dimensional vector space V over the commonscalar field F . In this setting it is more natural to study additivebilinear forms ( | ) : V × V → ( F, +) rather than multiplicative bilinearforms (cid:104)|(cid:105) : V × V → ( T, · ) for a torus T . The latter kind is our exclusiveobject of study in the present paper, but for contrast we shall refer tothem in this context by their alternative name bicharacter . Fixing any character χ : ( F, +) → ( T, · ) , one obtains the associated bicharacter bysetting (cid:104) ξ | x (cid:105) := χ ( ξ | x ) for ( x, ξ ) ∈ V × V . By a character we meanhere a group homomorphism that we may take to be an epimorphism(shrinking the torus T if necessary).The bicharacter (cid:104)|(cid:105) induced by the additive bilinear form ( | ) is clearlydegenerate if the latter is. As noted above, the converse is true if werestrict ourselves to standard characters . We define these as epimor-phisms χ : ( F, +) → ( T, · ) such that ker χ = Z ∗ , where Z ∗ denotesthe “prime ring” of F , meaning the smallest nontrivial subring of F .We have Z ∗ = Z p if the field F has positive characteristic p , other-wise Z ∗ = Z . The resulting duality (cid:104) V | V (cid:105) will be called an abstractvector duality . // Using standard characters, nondegeneracy may be viewed equiva-lently as a property of bilinear forms or their bicharacters. In thesequel, we may therefore identify nondegenerate bilinear forms withtheir associated bicharacters.
Lemma 8.
For a bilinear form ( | ) : V × V → F on a vector space,let (cid:104)|(cid:105) : V × V → T be its associated bicharacter with respect to any ∗ fixed standard character χ : ( F, +) → ( T, · ) . Then (cid:104)|(cid:105) is nondegenerateiff ( | ) is nondegenerate.Proof. See Lemma (cid:74) ?? (cid:75) . (cid:3) Example 9.
Typically, the bilinear form on V = F n is the standard in-ner product (cid:104) ξ | x (cid:105) = ξ · x = ξ x + · · · + ξ n x n , yielding the n -dimensionalvector duality over F . Since the inner product is also equivalent to thenatural paring F n × F n → F , one may use the dot notation for both.The most important special case of Example 7 is given by the com-plex field K = C with the classical torus T = T and vector spaces V = R n having their canonical Euclidean structure. Via the standard char-acter χ ( t ) = e iτt , one obtains the bicharacter (cid:104) ξ | x (cid:105) = e iτx · ξ . Weshall refer to this example as the n -dimensional standard vector du-ality (cid:104) R n | R n (cid:105) , which may also be presented as (cid:104) R n | R n (cid:105) under thenatural pairing. // Example 10. If V = F n is a vector space over a Galois field F = GF( q ) with q = p m ( m ∈ N ) elements, we may again use the usual dot productas a bilinear form. Using the multiplicative cyclic group (cid:104) ζ p (cid:105) ⊂ Q ( ζ p ) generated by a p -th unit root ζ p as in the proof of Lemma 8, onegets a standard character χ a : GF( q ) → (cid:104) ζ p (cid:105) , c (cid:55)→ ζ tr acp , for arbitrary a ∈ Z × p and trace map tr : GF( q ) → Z p , c (cid:55)→ c + c p + · · · + c p m − ;for details regarding characters on Galois fields see [17]. With theinduced duality (cid:104) ξ | x (cid:105) = χ a ( x · ξ ) , we call this example the modularvector duality (cid:104) GF( q ) n | GF( q ) n (cid:105) . // As is well known, the torus duality gives rise to Fourier series (seeExample 40b) and the vector duality to Fourier integrals (see Exam-ple 40a). There is another well-known breed of Fourier transforms,known as the discrete Fourier transform (see Example 40d); it is basedon the following important duality.
Example 11.
Given any N ∈ N , let us introduce two concrete real-izations of the cyclic group of order N . The first is the common repre-sentation Z N := { , , . . . , N − } with addition modulo N , the secondis T N := N − Z N = { , N , . . . , N − N } with addition modulo . The du-ality T nN × Z nN → T is now defined by (cid:104) ξ | x (cid:105) := e iτx · ξ . This can be ob-tained from the torus duality in two steps: First one restricts to the bi-linear form (cid:104)|(cid:105) : T nN × Z n → T via the embedding T nN (cid:44) → T n , ξ i (cid:55)→ e iτξ i .Then one applies the usual universal construction [62, §I.9] for mak-ing bilinear forms into dualities by taking the quotient on the rightmodulo the right kernel ( N Z ) n ; nothing is needed on the left sincethe left kernel is trivial. The resulting duality will be called the cyclicduality (cid:104) T nN | Z nN (cid:105) . N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 15
Of course, one may also form the conjugate cyclic duality (cid:104) Z nN | T nN (cid:105) .But unlike their infinite relatives, the dualities (cid:104) T nN | Z nN (cid:105) and (cid:104) Z nN | T nN (cid:105) are not only similar but actually the same (i.e. isomorphic): Since thecyclic groups T N and Z N are the same abstract group Z /N , both areone and the same duality (cid:104) ( Z /N ) n | ( Z /N ) n (cid:105) , given [37, Thm. 4.5d] by(7) (cid:104) k + N Z n | l + N Z n (cid:105) = e iτ ( k · l ) /N for k, l ∈ Z n . Nevertheless, it can be worthwhile to distinguish thetwo realizations of this duality as we shall see when introducing thecorresponding Fourier operators in Example 40d.The different nature of T nN and Z nN can also be seen in the context of normalizing Haar measure µ . While such a choice is per se immaterial,it must be consistent between the position and momentum group forthe inversion theorem to hold [100, §1.5.1]. For compact groups G , thecanonical choice is to set µ ( G ) = 1 , while for discrete groups G onesets µ ( { x } ) = 1 for all points x ∈ G . But since ( Z /N ) n happens tobe both discrete and compact, one must decide whether to impose thediscrete or the compact normalization on the position group ( Z /N ) n ,so that the other normalization is then conferred onto its dual, which isagain ( Z /N ) n . From the above construction, it is clear that the naturalchoice is to endow T nN with the compact and Z nN with the discretenormalization. In other words, we have µ ( T nN ) = 1 and µ ( Z nN ) = N n ,thus also µ ( { x } ) = 1 /N n for x ∈ T nN but µ ( { x } ) = 1 for x ∈ Z nN . // Example 12.
Following the nice approach layed out in [79], one cangeneralize the cyclic duality from the group Z N to an arbitrary finitegroup G . If the latter has order n and exponent m , one may take forthe torus any integral domain R that contains /n and a primitive m -th root of unity. These two conditions are shown to be sufficient andnecessary for R [ G ] ∼ = R n , which will be relevant for the correspondingFourier transform sketched in Example 42 below. Note that one mayalways take R = T , as is done for the cyclic duality.Writing R ∗ for the multiplicative group of R , the above conditionsimply that | ˆ G | = n for the “dual group” ˆ G := Hom( G, R ∗ ) and thatthe bilinear form ν : ˆ G × G → R ∗ with (cid:104) ξ | x (cid:105) ν := ξ ( x ) is a duality;confer [79, (3.10)]. We will refer to ν as the Nicholson duality of thegroup G over the ring R . // Before concluding with the last crucial class of examples for Heisen-berg groups, let us briefly mention a rather degenerate example thatis nevertheless occasionally useful when studying (counter)examples ofvarious properties that may be ascribed to Heisenberg groups. ∗ Example 13. If R is a ring, its multiplication map may be viewed asa bilinear map · : R + × R + → R + , where R + is the additive group of R .As a bilinear form, · : R + × R + → R + is degenerate iff Ann R ( R ) (cid:54) = 0 .In that case we call (cid:104) R + | R + (cid:105) the symmetric duality of the ring R ,with associated Heisenberg group H ( R ) := R + R + (cid:111) R + . If one iswilling to generalize the setting of Example 7 from F -vector spaces to R -modules, the symmetric duality of R is the “vector duality” on theone-dimensional module R under the trivial (non-standard) character χ : R + → R + , x (cid:55)→ x . // The one-dimensional cyclic duality T N × Z N → T is the multi-plication map Z N × Z N → Z N , transported from the additive valuegroup Z N ∼ = N − Z N = { , N , . . . , N − N } ⊂ R into the multiplicativegroup T via the standard character χ : R → T , t (cid:55)→ e iτt of Exam-ple 9. It is interesting to see what one gets for the plain multiplicationmap Z N × Z N → Z N . In the following example, we explore this ques-tion for the special case N = 2 . Example 14.
So we appply the construction R + R + (cid:111) R + to the ring R = Z , corresponding to the duality β : Z × Z → Z defined by β ( m, n ) = mn . True to our conventions, we write the torus multiplica-tively via Z ∼ −→ Z × , [ c ] (cid:55)→ ( − c , meaning [0] ↔ +1 , [1] ↔ − . Asusual, we often express the values ± just by the sign. The dualityis then given by β ( m, n ) = ( − mn . We will show that H ( β ) is thedihedral group D , the symmetry group of the square (which we as-sume centered in the origin with axis-paraellel sides). If t denotes thecounter-clockwise ◦ turn and r the reflection in the vertical axis, weobtain the presentation D = (cid:104) t, r | t = r = 1 , rt = t r (cid:105) = { , t, t , t , r, tr, t r, t r } , Where tr, t r, t r may be respectively interpreted as reflections in theanti-diagonal x + y = 0 , horizontal y = 0 , and diagonal x − y = 0 .We choose Z ( D ) = { , t } as our torus T = Z , which enforces theidentification ↔ +1 , t ↔ − . For the position group G = Z we take { , r } , leading to the identification ↔ [0] , r ↔ [1] ; for themomentum group Γ = Z we use { , tr } with identification ↔ [0] , tr ↔ [1] . In these terms, the duality β : Z × Z → Z , ( m, n ) (cid:55)→ mn sends ( r, tr ) to − and all other pairs to +1 . D H ( β ) D H ( β )1 +00 r +10 t − tr +01 t − t r − t +11 t r − N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 17
For defining a group isomorphism D ∼ −→ H ( β ) , we construct first theunique homomorphism on the free group with (cid:55)→ +00 , t (cid:55)→ − , r (cid:55)→ +10 . As one sees immediately, this homomorphism annihilatesthe relators t , r , trtr and thus yields a homomorphism ι : D → H ( β ) .Computing all other elements in terms of the generators t and r , onewill verify that ι is given as in the table above. Since this is obviouslya bijection, it provides us with the required isomorphism D ∼ −→ H ( β ) . // For the standard vector duality (cid:104) R n | R n (cid:105) of Example 9, the link to thematrix group (6) can be made more precise by reframing the Heisen-berg group in terms of a symplectic vector space V . This is the formu-lation commonly used in more advanced treatments of the Heisenberggroup [18, §5.1], [38], [65]. Example 15.
As we have seen above (Lemma 3), all Heisenberg ex-tensions associated with a fixed commutator form can be representedby the so-called symplectic Heisenberg group, provided the torus isuniquely -divisible. The most important instance arises for a finite-dimensional symplectic vector space ( Z, Ω) over a field F . Here onetakes the additive group ( F, +) as torus and Ω : Z × Z → F as com-mutator form to yield the symplectic Heisenberg group [ H ] Ω / via thenilquadratic extension F (cid:26) [ H ] Ω / (cid:16) Z . Note that we write here Ω / for the “square root” of the commutator form since the torus is writtenadditively. Taking Z = R n , this agrees with [38, p. 19], where [ H ] Ω is written H n . Dissociated from the underyling symplectic structure,the factor / in [ H ] Ω / may look arbitrary (it can obviously be elimi-nated by rescaling), but it also ensures that exponentiation yields thecanonical commutators of Hamiltonian mechanics [38, (1.15)].We have already mentioned earlier that the symplectic Heisenberggroup is not a Heisenberg group stricto sensu (as per Definition 1).But it is closely related to a whole bunch of them via polarization . Fora symplectic vector space ( Z, Ω) , a polarization by itself is just a choiceof Lagrangian subspace G ≤ Z , so that a Lagrangian splitting ( G, Γ) of Z could be viewed as a “duplex polarization” (noting that any La-grangian bisection is a splitting for vector spaces). The polarizations G make up the so-called Lagrange-Grassmann manifold, where each G hasplenty of Lagrangian complements Γ ; see [65, Prop. A6.1.6]. Any choiceof Lagrangian splitting ( G, Γ) induces a natural symplectic isomor-phism ι G, Γ : Z ∼ −→ G ⊕ G ∗ fixing G and sending ξ ∈ Γ to Ω( ξ, − ) ∈ G ∗ ;here injectivity follows from the Lagrangian nature of G while surjectiv-ity needs the finite dimensionality of Z . Here G ⊕ G ∗ has the canonicalsymplectic structure given by Ω G ( x, ξ ; ˜ x, ˜ ξ ) = ( ξ | ˜ x ) − ( ˜ ξ | x ) , where ( −|− ) ∗ denotes the natural pairing on G ∗ × G . Via ι G, Γ , symplectic bases of Z are in bijective correspondence with bases of G .As mentioned after Definition 2, each Lagrangian splitting ( G, Γ) of Z yields a unique Heisenberg group H ( β Ω ) ; its associated bilinearform β Ω : Γ × G → F is given by β Ω ( ξ, x ) = ( ξ | x ) . In the standardcase Z = R n , the Heisenberg group H ( β Ω ) agrees with H pol n in [38,p. 19]. In this case, the matrix multiplication law (6) applies literally,giving rise to the well-known matrix Lie group. By Lemma 3, it is iso-morphic to the symplectic Heisenberg group via the polarization map P G, Γ : [ H ] Ω ∼ −→ H ( β Ω ) with ( λ ; x, ξ ) (cid:55)→ ( λ + ( ξ | x ); x, ξ ) .If a standard character χ : ( F, +) → ( T, · ) is available, one can con-struct yet another version of both [ H ] Ω and H ( β Ω ) by pushing forwardthe F -valued commutator form Ω to its T -valued cousin ω := χ ◦ Ω .Even if T is not (uniquely) -divisible, the commutator form ω has a standard square root , namely √ ω = χ ◦ ◦ Ω , where : F → F is divi-sion by two (so the torus T is supposed to have characteristic differentfrom ). In the classical setting F = C , the torus T = T ⊂ C is -divisible but not uniquely so; one may here use the standard character χ ( t ) = e iτt , which is tantamount to choosing the principal branch ofthe multivalued square-root function.In fact, the strategy of forging square roots from the halving iso-morphism : F → F may be generalized as follows by (cid:74) ?? (cid:75) . If onehas an (additively written) uniquely -divisible torus ( ˆ T , +) above thetorus ( T, · ) via χ : ˆ T (cid:16) T , the pushforward χ ∗ : Ω ( P, ˆ T ) → Ω ( P, T ) induces a section of the skewing map q in the short exact sequence (2)over Ω χ ( P, T ) := im( χ ∗ ) ≤ Ω ( P, T ) , namely the map χ ∗ (Ω) (cid:55)→ χ ∗ (Ω / .Lemma 3 is generalized in Proposition (cid:74) ?? (cid:75) , yielding an isomorphism [ H ] ω ∼ −→ H ( β ) for ω ∈ Ω χ ( P, T ) .Back to vector spaces, we can now construct the little symplecticHeisenberg group [ H ] ω = T × √ ω Z , along with the corresponding littleHeisenberg group H ( β ω ) . The latter has the bilinear form β ω = χ ◦ Ω ,which reads (cid:104) x | ξ (cid:105) = χ ( x | ξ ) in our usual bracket notation. In terms ofthe projections π Γ : Z → Γ and π G : Z → G , we have β ω = ω ◦ ( π Γ × π G ) as well as β Ω = Ω ◦ ( π Γ × π G ) for the big symplectic Heisenberg group [ H ] Ω = F × Ω / Z . From the generalized Lemma 3 we obtain the littlepolarization map p G, Γ : [ H ] ω ∼ −→ H ( β ω ) with p G, Γ ( c ; x, ξ ) = c (cid:112) (cid:104) ξ | x (cid:105) .The canonical epimorphism π Z := χ × Z : [ H ] Ω (cid:16) [ H ] ω has kernel ker( χ ) × ∼ = Z ∗ , so the little symplectic Heisenberg group is the bigone modulo the prime ring; a similar statement holds for the Heisenberggroups via the (set-theoretically) same map π Z : H ( β Ω ) → H ( β ω ) . N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 19 [ H ] Ω P G, Γ (cid:47) (cid:47) π Z (cid:15) (cid:15) H ( β Ω ) π Z (cid:15) (cid:15) [ H ] ω p G, Γ (cid:47) (cid:47) H ( β ω ) Figure 1.
Symplectic Heisenberg GroupThe “Heisenberg group” given in [1, Exc. 5.1-4] is essentially [ H ] ω forthe symplectic space Z = R n with standard character χ : R → T asabove, except that they shun the factor / so effectively rescale Ω bythe factor . As a consequence of this, they incur the factor in thecanonical commutator [1, Exc. 5.1-4b].We have given references for all but one of the Heisenberg flavors inFigure 1. The remaining group H ( β ω ) coincides with the Heisenberggroup of the abstract vector duality (cid:104) V | V (cid:105) for V = G = Γ and thechosen standard character χ : F → T , where as before ( ξ | x ) = β Ω ( ξ, x ) with β Ω = Ω ◦ ( π Γ × π G ) . The general case of vector dualities is treatedin Example 7. In the important special case V = R n , this yields thestandard vector duality (cid:104) R n | R n (cid:105) or isomorphically (cid:104) Γ | G (cid:105) = (cid:104) R n | R n (cid:105) ;see Example 9. It is this case which leads to the famous classical Fourierintegral, as we shall see in Example 40a below. // Remark 16.
Let us add a few remarks on the physical interpreta-tion of Example 15. The symplectic vector space T ∗ V = V ⊕ V ∗ isnothing but the Hamiltonian phase [73, §1.1], whereas the abstractvector duality (cid:104) V | V (cid:105) of Example 7 is linked to the Lagrangian phasespace T V = V ⊕ V . In both cases, the elements of V denote posi-tions while the tangent/cotangent vectors are the velocities/momenta .This generalizes to the nonlinear case where the configuration space isa manifold M rather than a vector space V , with Lagrangian phasespace T M and Hamiltonian phase space T ∗ M .The Hamiltonian case is important since it leads to quantization viareplacing the commutative algebra of classical observables C ∞ ( T ∗ V ) by the noncommutative algebra (cid:72) (cid:0) L ( V ) (cid:1) of self-adjoint Hermitianoperators on the Hilbert space L ( V ) . The observables position andmomentum are quantized [43, §3.5] as position operator f ( x ) (cid:55)→ x f ( x ) and canonically conjugate momentum operator f ( x ) (cid:55)→ ∂f /∂x ). Thisis intimately linked to the unitary irrep (= irreducible representation)of the Heisenberg group (see Remark 55 below). (cid:125) In closing this short investigation of Heisenberg groups, let us cor-roborate our earlier claim about the failure of existence/uniqueness ∗ of abelian splittings for general nilquadratic groups. Regarding exis-tence, we refer to Example (cid:74) ?? (cid:75) , which makes it clear that the freenilquadratic group N fails to have an abelian splitting. Example 17.
For seeing that one and the same nilquadratic groupmay be equipped with different abelian splittings , take the standardvector duality β = (cid:104) R | R (cid:105) of Example 9. The corresponding Heisen-berg group H ( β ) = TR (cid:111) R with the classical torus T ⊂ C × andphase space R comes endowed with the standard Lagrangian split-ting ( R × , × R ) . But it is easy to see that any other direct decom-position G (cid:117) Γ = R with one-dimensional subspaces G, Γ ≤ R alsoyields a Lagrangian splitting ( G, Γ) , which is however distinct fromthe standard one. Since Lagrangian splittings of P are in bijectivecorrespondence with abelian splittings of H by Theorem (cid:74) ?? (cid:75) , this es-tablishes that the former are not uniquely determined for the givenHeisenberg extension T ι (cid:26) H π (cid:16) R . // The Category of Heisenberg Algebras.
Analysis deals withFourier operators on real or complex function spaces, which are en-dowed with certain
Heisenberg actions (which we define here as linearrepresentations of Heisenberg groups). Our task in this subsection isto capture this idea in a suitable algebraic setting.A Heisenberg group H ( β ) typically comes with an action on somefunction space on which the torus T ≤ H ( β ) “acts naturally via scalars”(see Example 40 below for classical cases). For making this precise, weview the function spaces as modules or algebras over a fixed scalarring K , and the latter equipped with a torus action ∗ : T → Aut K ( K ) .Thus we have the action laws ∗ λ = λ and ( cd ) ∗ λ = c ∗ ( d ∗ λ ) as wellas linearity c ∗ ( λ + µ ) = c ∗ λ + c ∗ µ and c ∗ ( λµ ) = ( c ∗ λ ) µ = λ ( c ∗ µ ) ,for all c, d ∈ T and λ, µ ∈ K . Equivalently, the torus action ∗ may bedescribed by the map ε T : ( T, · ) → ( K × , · ) with ε T ( c ) := c ∗ K sincewe have c ∗ λ = ε T ( c ) λ . In the sequel, we shall refer to both ∗ and ε T as a torus action.Any K -module S is naturally a T -module since, if ∆ : K → End Z ( S ) is the structure map of S with induced action ∆ × : K × → Aut K ( S ) , weobtain an action ∆ T of T on the abelian group ( S, +) by ∆ T = ∆ × ◦ ε T .We can now give a precise meaning to the informal phrase used in theprevious paragraph: If η : H ( β ) → Aut K ( S ) is any K -linear action, wesay the torus acts naturally via ε T if η factors through ∆ T . In otherwords, we require the following diagram to commute: N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 21 T ε T (cid:47) (cid:47) (cid:127) (cid:95) (cid:15) (cid:15) K × ∆ × (cid:15) (cid:15) H ( β ) η (cid:47) (cid:47) Aut K ( S ) If we have a Heisenberg action on a K -algebra ( S, + , (cid:5) ) , we shallalways designate the action of u ∈ H ( β ) on an element s ∈ S by u • s ,thus avoiding confusion with the pointwise product · to be introcuedlater. In relation to Fourier structures, we encounter two cruciallydistinct flavors of elements in H ( β ) in relation to the multiplicativestructure of S : • We call u ∈ H ( β ) a Heisenberg scalar if u • ( s (cid:5) ˜ s ) = ( u • s ) (cid:5) ˜ s for all s, ˜ s ∈ S . • We call u ∈ H ( β ) a Heisenberg operator if u • ( s (cid:5) ˜ s ) = ( u • s ) (cid:5) ( u • ˜ s ) for all s, ˜ s ∈ S .When the torus acts naturally via ε T , the torus elements are clearlyHeisenberg scalars.We are now in a position to give a concise algebraic description of thefunction spaces underlying Fourier transforms. Following the terminol-ogy of [115] and [87], we call such spaces Heisenberg modules since theyare modules over some Heisenberg group H ( β ) . In a natural—thoughless conventional—extension, we shall speak of a Heisenberg algebra ifthe module is additionally equipped with a compatible multiplication. Definition 18.
Let β : Γ × G → T be a duality, and let K be a ringwith torus action ε T . • A Heisenberg module over β is a K -module S with a linearaction H ( β ) × S → S where the torus acts naturally via ε T . • A Heisenberg algebra over β is a K -algebra and a Heisenbergmodule where all elements of T G ≤ H ( β ) are Heisenberg scalarswhile all elements of Γ ≤ H ( β ) are Heisenberg operators.A Heisenberg morphism from a Heisenberg module/algebra S over β to another Heisenberg module/algebra S (cid:48) over β is defined to be anequivariant K -module/algebra homomorphism S → S (cid:48) . Later on, we will consider mainly two multiplicative structures given defined bya Pontryagin duality—the convolution (cid:63) and the pointwise product · just mentioned. This should not be confused with the classical Lie algebra of H n ( R ) , prop-erly [38, p. 18] called “Heisenberg Lie algebra”, but occasionally [38, p. 18, 21]shortened to “Heisenberg algebra”. ∗ For avoiding tedious repetitions, we shall from now on regard
Heisen-berg modules as degenerate Heisenberg algebras , in the sense of havingtrivial multiplication. We extend also the scalar/operator terminologyto Heisenberg groups H ( β ) per se, regardless of any Heisenbeg alge-bras on which they might act (dropping the qualification “Heisenberg”where it is obvious): Elements of T G ≤ H ( β ) will be called scalars ,elements of Γ ≤ H ( β ) operators , and generic elements actors . (Thiswill not lead to any confusion since we shall never encounter algebrasthat are Heisenberg modules but not Heisenberg algebras.)Let us spell out the definition of Heisenberg algebras in more detail,splitting the Heisenberg action into the three actions G × S → S and Γ × S → S and T × S → S , while writing multiplication in the algebra asjuxtaposition. In addition to the K -algebra axioms, for arbitrary s, s (cid:48) ∈ S and c ∈ T and x, x (cid:48) ∈ G and ξ, ξ (cid:48) ∈ Γ , we require the followingidentities :(H ) G • s = s (H ) Γ • s = s (H ) ( xx (cid:48) ) • s = x • ( x (cid:48) • s ) (H ) ( ξξ (cid:48) ) • s = ξ • ( ξ (cid:48) • s ) (H ) c • s = ε T ( c ) s (H ) ξ • ( x • s ) = (cid:104) ξ | x (cid:105) x • ( ξ • s ) (H ) x • ( ss (cid:48) ) = ( x • s ) s (cid:48) (H ) ξ • ( ss (cid:48) ) = ( ξ • s )( ξ • s (cid:48) ) Dropping the last two axioms (and requiring S to be a K -moduleinstead of a K -algebra), one obtains a Heisenberg module instead of aHeisenberg algebra over β . It is easy to check that the above identitiesare equivalent to the requirements of Definition 18. For example, the twist axiom (H ) is a consequence of the composition law in H ( β ) . TheHeisenberg action may be decomposed into the “scalar action” of T G and the “operator action” of Γ ; taking this view, S is a twisted bimoduleunder (H ). On another view, Heisenberg modules over β are linearrepresentations of H ( β ) , with Heisenberg morphisms as intertwiners.At any rate, the category of Heisenberg modules over β is denotedby ModH ( β ) , and the category of Heisenberg algebras by AlgH ( β ) . Byour convention of regarding Heisenberg modules as degnerate Heisen-berg algebras, ModH ( β ) is a full subcategory of AlgH ( β ) . The assign-ments β (cid:55)→ ModH ( β ) and β (cid:55)→ AlgH ( β ) may be seen as contravariantfunctors ModH : Du → Cat and
AlgH : Du → Cat in the followingway: If ( g × γ, t ) is a morphism between dualities β : G × Γ → T and β (cid:48) : G (cid:48) × Γ (cid:48) → T (cid:48) , the functor AlgH ( g × γ, t ) : AlgH ( β (cid:48) ) → AlgH ( β ) is defined as follows. N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 23 • On objects, it acts by sending S (cid:48) ∈ AlgH ( β (cid:48) ) to the twistedmodule S := S (cid:48) [ σ ] ∈ AlgH ( β ) , with the twist map givenby σ := H ( g × γ, t ) : H ( β ) → H ( β (cid:48) ) . • On morphisms, the functor
AlgH ( g × γ, t ) acts trivially: Amorphism ϕ (cid:48) : S (cid:48) → S (cid:48) in AlgH ( β (cid:48) ) stays the same since ϕ = ϕ (cid:48) respects the σ -twisted action of H ( β ) .It is easy to see that AlgH : Du → Cat is then indeed a contravariantfunctor, and the same holds for
ModH : Du → Cat .By the well-known Grothendieck construction [14, §12.2], the functor
AlgH : Du → Cat yields a split fibration π : AlgH (cid:111) Du → Du ,where the category AlgH (cid:111) Du has the objects ( β, S ) with β ∈ Du , S ∈ AlgH ( β ) , and morphisms ( ϕ, Φ) : ( β, S ) → ( β (cid:48) , S (cid:48) ) with ϕ : β → β (cid:48) in Du and Φ : S → AlgH ( ϕ ) S (cid:48) in AlgH ( β ) . Writing ϕ = ( g × γ, t ) forsuitable morphisms g × γ : G × Γ → G (cid:48) × Γ (cid:48) and t : T → T (cid:48) , this requiresthe conditions Φ( x • s ) = g ( x ) • Φ( s ) and Φ( ξ • s ) = γ ( ξ ) • Φ( s ) as wellas Φ( c • s ) = t ( c ) • Φ( s ) for s ∈ S and c ( x, ξ ) ∈ H ( β ) . Composition ofmorphisms is defined by ( ϕ , Φ ) ◦ ( ϕ , Φ ) = ( ϕ ◦ ϕ , AlgH ( ϕ ) Φ ◦ Φ ) , as usual for semidirect products. As a result, the fibered category AlgH ( ) :=
AlgH (cid:111) Du of Heisenberg algebras is a disjoint union AlgH ( ) = (cid:93) β ∈ Du AlgH ( β ) similar to the well-known fibration Alg = (cid:85) R ∈ Rng
Alg R .As noted above, AlgH ( β ) includes ModH ( β ) . It is clear that thecategory ModH ( ) :=
ModH (cid:111) Du of Heisenberg modules has ananalogous fibration over Du , similar to the fibration of Mod over
Rng .The category
AlgH ( β ) has products , namely the direct product of K -algebras with componentwise Heisenberg action. The commutativediagram for products carries over from Alg K to AlgH ( β ) since theprojection maps π : S × S → S and π : S × S → S are Heisenbergmorphisms.The category AlgH ( β ) also has a tensor product . Given Heisenbergalgebras S, S (cid:48) ∈ AlgH ( β ) , we view them as KG -algebras and endow S ⊗ KG S (cid:48) ∈ Alg KG with an action of H ( β ) = T G (cid:111) Γ as follows: Theaction of T is via ε T , that of G via the KG -module structure, whileHeisenberg operators ξ ∈ Γ act via ξ • s ⊗ s (cid:48) := ( ξ • s ) ⊗ ( ξ • s (cid:48) ) . ∗ One checks immediately that we have in fact S ⊗ KG S (cid:48) ∈ AlgH ( β ) .For brevity, we refer to this Heisenberg algebra as S ⊗ S (cid:48) . Theorem 19.
Let β be any duality. Then AlgH ( β ) is a symmetricmonoidal category.Proof. It is clear that
AlgH ( β ) is a monoidal category since the usualassociators α ABC : ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ) as well as the unitors λ A : ( KG ) ⊗ A → A and ρ A : A ⊗ ( KG ) → A of the category Alg KG areeasily seen to be Heisenberg isomorphisms. For the unitors, it shouldbe noted that KG is a Heisenberg algebra with action defined via ε T and ξ • y := (cid:104) y | ξ (cid:105) y . Finally, AlgH ( β ) is a symmetric monoidal categorysince the braiding γ AB : A ⊗ B → B ⊗ A of Alg KG is also a Heisenbergisomorphism. (cid:3) Unlike in
Alg K , the tensor product in AlgH ( β ) is not a coproduct ,however. The problem is that Heisenberg algebras are typically withoutunit element so that the tensor product generally lacks the injectionmaps S → S ⊗ S (cid:48) and S (cid:48) → S ⊗ S (cid:48) required of coproducts.This may be remedied by taking recourse to the coproduct of com-mutative nonunital algebras . The construction is straightforward butdifficult to locate in the literature. If A and B are commutative nonuni-tal algebras over a commutative unital ring R , their coproduct is de-fined by A ˆ ⊗ R B := A ⊕ B ⊕ ( A ⊗ R B ) with multiplication given bysetting ( a , b , a (cid:48) ⊗ b (cid:48) ) · ( a , b , a (cid:48) ⊗ b (cid:48) ) equal to ( a a , b b , a ⊗ b + a ⊗ b + a a (cid:48) ⊗ b (cid:48) + a (cid:48) a ⊗ b (cid:48) + a (cid:48) ⊗ b b (cid:48) + a (cid:48) ⊗ b (cid:48) b + a (cid:48) a (cid:48) ⊗ b (cid:48) b (cid:48) ) . Then we have the coproduct injections ι : A → A ˆ ⊗ R B , a (cid:55)→ ( a, , and ι : B → A ˆ ⊗ R B , b (cid:55)→ (0 , b, .Using this structure, it is straightforward to define the coproductof S, S (cid:48) ∈ AlgH ( β ) . Viewing S and S (cid:48) again as KG -algebras, weequip S ˆ ⊗ KG S (cid:48) = S ⊕ S (cid:48) ⊕ ( S ⊗ KG S (cid:48) ) with the componentwise Heisen-berg action induced by those on S , S (cid:48) and S ⊗ KG S (cid:48) , where the latteris the Heisenberg module S ⊗ S (cid:48) introduced above. It is easy to seethat ι : S → S ˆ ⊗ KG S (cid:48) and ι : S (cid:48) → S ˆ ⊗ KG S (cid:48) are Heisenberg mor-phisms, hence S ˆ ⊗ KG S (cid:48) is indeed a coproduct in AlgH ( β ) , which wealso abbreviate by S ˆ ⊗ S (cid:48) .A Heisenberg algebra is a Heisenberg module whose structure is en-riched by tacking on a nontrivial product (replacing the trivial defaultproduct). In Fourier theory (see §4), one encounters modules withan even richer structure—modules that carry two products. To makethings precise, let us call ( A, (cid:63), · ) a twain algebra over the commutative N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 25 and unital ring K if A is a K -module with two bilinear products (cid:63) and · . The corresponding algebras ( A, (cid:63) ) and ( A, · ) will be denotedby A (cid:63) and A • . If one of the products is trivial, we have a plain al-gebra —if both are trivial, we retain the naked K -module, which wemight then call a slain algebra . Clearly, there are corresponding no-tions of twain/plain/slain homomorphisms, depending on how manymultiplication maps need to be preserved.A twain algebra over K may be described by giving two K -algebras ( A, (cid:63) ) and ( B, · ) together with a K -linear isomorphism ι : A ∼ −→ B .This yields the twain algebra ( A, (cid:63), · ) , where we write its transferredproduct x · y := ι − ( ιx · ιy ) for x, y ∈ A with the same symbol. Insuch a case we shall use the notation A (cid:62) ι B for the correspondingtwain algebra (suppressing the subscript ι when the isomorphism isunderstood), which we call the overlay of A on top of B .We shall also employ the casual twain/plain/slain jargon in conjunc-tion with Heisenberg actions of H ( β ) = T G (cid:111) Γ . To this end, let usrecall that a Heisenberg module over β was called a Heisenberg algebraover β if it is endowed with a bilinear multiplication such that the po-sitions x ∈ G ≤ H ( β ) act as scalars while the momenta ξ ∈ Γ ≤ H ( β ) act as operators (of course the c ∈ T ≤ H ( β ) always act as scalars).Borrowing typographic terminology, we shall also call such a structurea recto Heisenberg algebra . In contrast, a Heisenberg module over β will be called a verso Heisenberg algebra if it has a bilinear multipli-cation where the positions are operators and the moments scalars. (Inthe sequel, the qualification “recto” will only be used for emphasis orsymmetry.)Now a twain algebra A is called a Heisenberg twain algebra if A (cid:63) isa recto Heisenberg algebra while A • is a verso Heisenberg algebra. Ofcourse, the terms Heisenberg plain algebra and
Heisenberg slain algebra are just synonyms for “Heisenberg module” and “Heisenberg algebra”,respectively. This somewhat flippant terminology will come in handywhen dealing with Fourier structures. Note also that a Heisenbergtwain morphism is just a Heisenberg morphism that is at the sametime a twain homomorphism. We denote the category of Heisenbergtwain algebras over β by AlgH ( β ) .Let us now look at a rather simple specimen of a Heisenberg twainalgebra, here formulated in entirely algebraic terms. Its analytic signif-icance as a Fourier structure will be recognized in Example 53c below. Example 20.
The toroidal twain algebra is defined as C [ Z n ] (cid:62) C (cid:76) Z n (cid:77) ,where (cid:0) C [ Z n ] , (cid:63) (cid:1) denotes the group algebra (with Z n the free abeliangroup on n generators) while (cid:0) C (cid:76) Z n (cid:77) , · (cid:1) is the path algebra (with Z n ∗ viewed as a discrete quiver). In other words, the two products aredefined on the generators z a ( a ∈ Z n ) as z a (cid:63) z a (cid:48) = z a + a (cid:48) , z a · z a (cid:48) = δ a,a (cid:48) z a . Thus C [ Z n ] is the Laurent polynomial algebra while C (cid:76) Z n (cid:77) is the com-plex algebra generated by the orthogonal idempotents z a . ViewingLaurent polynomials (cid:80) a c a z a as multivariate sequences ( c a ) a ∈ Z n , theirproduct (cid:63) is the discrete convolution (see also Example 40c below).Let (cid:104)|(cid:105) Tor : T n × Z n → T be the torus duality (cid:104) ξ | x (cid:105) Tor = ξ x introducedin Example 6 with Heisenberg group H ( Tor ) = TZ n (cid:111) T n . We definethe Heisenberg action H ( Tor ) × C [ Z n ] → C [ Z n ] via x · z a = z a + x , ξ · z a = ξ a z a for ( x, ξ ) ∈ Z n × T n , and where the torus T ⊂ C acts trivially. It iseasy to see that ( C [ Z n ] , (cid:63), · ) is then a Heisenberg twain algebra. Forlater reference, let us also note the interchange law (8) ( z a · z b ) (cid:63) ( z c · z d ) = δ a + d,b + c ( z a (cid:63) z c ) · ( z b (cid:63) z d ) , which is an immediate consequence of the composition laws given above.Used from left to right, its effect is similar to the distributivity axiom.The induced normal form of ( (cid:63), · ) terms is then a complex linear com-bination of pointwise products of convolutions. // The Heisenberg Twist.
We now turn our attention to an inter-esting feature of Heisenberg groups that is also important for under-standing the nature of Fourier operators, in particular when iterated.Up to now we have been speaking of left
Heisenberg algebras, omit-ting the qualification “left” since we have not yet considered their right counterparts. For introducing Fourier operators, though, the distinc-tion between left and right Heisenberg algebras turns out to be crucial.Referring to Definition 18, a right Heisenberg algebra S is exactly thesame except that one has a linear right action S × H ( β ) → S . Thelatter induces a map η : H ( β ) o → Aut K ( S ) , which is again required tofactor through the restricted torus action ∆ T .In terms of axioms, a right Heisenberg action is also characterizedby (H )–(H ), except that the phase factor (cid:104) ξ | x (cid:105) in (H ) changes sides.Thus Heisenberg scalars u ∈ T G ≤ H ( β ) act as ( s (cid:63) ˜ s ) • u = s (cid:63) (˜ s • u ) ,Heisenberg operators u ∈ Γ as ( s (cid:63) ˜ s ) • u = ( s • u ) (cid:63) (˜ s • u ) . In thesequel, the unqualified term “Heisenberg algebra” is meant to refer toleft Heisenberg algebras, which we continue to denote by AlgH ( β ) .We recall the restriction of scalars for modules: Given a ring homo-morphism ϕ : R → S , any left S -module N can be turned into a left R -module ϕ ∗ ( N ) by precomposing its scalar action S → Aut Z ( N ) with ϕ . N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 27 If S is instead a right S -module with scalar action S o → Aut Z ( N ) , onegets a right R -module ϕ ∗ ( N ) by precomposing. We use the same def-inition and notation when ϕ is an antihomomorphism, but now thesides are swapped: If N is a left/right S -module, ϕ ∗ ( N ) is a right/left R -module.Let M be a left R -module, N a left/right S -module and ϕ : R → S ahomomorphism/antihomomorphism of rings. Then we call f : M → N a homomorphism over ϕ if f : M → ϕ ∗ ( N ) is a homomorphism of R -modules. Thus we have f ( λ • x ) = ϕ ( λ ) • f ( x ) in the homomorphicand f ( λ • x ) = f ( x ) • ϕ ( λ ) in the antihomomorphic case (adopting • forthe scalar action).The rings in question may be group rings Z H over arbitrary groups H .In that case, the modules are called H -modules in the terminology ofrepresentation theory, namely abelian groups on which H acts via au-tomorphisms [30, p. 95]. For such group modules, one often uses the in-version in H as a preferred antihomomorphism Z H → Z H for switchingbetween left and right H -modules. For Heisenberg groups H = H ( β ) ,however, two other antihomorphisms are more important—at least inthe context of Fourier analysis—and both are involutions (“twists”) inthe sense of involutive antihomomorphism . In addition, there is alsoan involutive homomorphism (“flip”), which plays an important role inFourier operators. Before investigating their signficance in some detail,let us first list their definitions:Forward Twist ˆ J : H ( β ) → H ( β ) o , c ( x, ξ ) (cid:55)→ c (cid:104) ξ | x (cid:105) ( x, − ξ ) (9) Backward Twist ˇ J : H ( β ) → H ( β ) o , c ( x, ξ ) (cid:55)→ c (cid:104) ξ | x (cid:105) ( − x, ξ ) (10) Parity Flip ¯ J : H ( β ) → H ( β ) , c ( x, ξ ) (cid:55)→ c ( − x, − ξ ) (11)Note that the twists ˆ J and ˇ J can be squared since each of them may alsobe viewed as an antihomomorphism H ( β ) o → H ( β ) , and they combineto the flip as ˆ J ˇ J = ¯ J = ˇ J ˆ J . One may picture the twist-and-flip actionsas a cyclic process of periodicity four, intimately linked with the actionof Fourier operators (see Figure 2 below and §3.3).For understanding the significance of those involutions, we remarkthat each duality β : Γ × G → T may be transposed to yield its mirrorimage β (cid:48) : G × Γ → T such that β (cid:48) ( x, ξ ) = β ( ξ, x ) . From the definitionof the Heisenberg group H ( β ) = T G (cid:111) Γ one sees that H ( β (cid:48) ) = T Γ (cid:111) G yields the dual Heisenberg group , which reverses the roles of positionsand momenta. The crucial fact to observe now is that H ( β (cid:48) ) ∼ = H ( β ) o via the isomorphism c ( ξ, x ) ↔ c ( x, ξ ) . For this reason we shall denotethe category of right Heisenberg algebras by AlgH ( β (cid:48) ) . ∗ Under the identification H ( β (cid:48) ) ∼ = H ( β ) o , the anti-isomorphism ˆ J be-comes an isomorphism J : H ( β ) → H ( β (cid:48) ) with c ( x, ξ ) (cid:55)→ c (cid:104) ξ | x (cid:105) ( − ξ, x ) .Projected onto the phase spaces P = G × Γ and P (cid:48) = Γ × G , this yieldsthe map j : P → P (cid:48) with ( x, ξ ) (cid:55)→ ( − ξ, x ) . This map is geometricallysignficant in at least two important examples:(a) If G = Γ is a finite-dimensional vector space over F , the phasespace P = V × V may be viewed as its complexification with j as its canonical complex structure. Taking ( | ) : V × V → F to bean additive bilinear form on V , the map j is like a rotation (it isliterally so if β is symmetric and positive definite over F = R ). Tak-ing V = R n , one recognizes j as the usual block matrix (cid:0) − I n I n (cid:1) ,so for n = 1 the action of j is multiplication by √− . Taking astandard character (Example 7), all this is seen to be an instanceof abstract vector duality .(b) In the case of the symplectic Heisenberg group H ( β Ω ) of Exam-ple 15, we have a finite-dimensional vector space G over F withdual Γ = G ∗ . The canonical symplectic form Ω G is then a skew-symmetric bilinear form on the phase space P = G × G ∗ . Inter-preted as a linear map into its dual, Ω G is just j : P → P ∗ ∼ = P (cid:48) ,and its matrix is again (cid:0) − I n I n (cid:1) .The second example can be generalized to arbitrary Heisenberg groups H ( β ) = T G (cid:111) Γ . It points to the right way of understanding the signif-icance of the tilt map j : P → P (cid:48) , namely as an alternative encoding ofthe symplectic structure expressed by ω . Indeed, taking the coproduct(as specified in Remark 4) of β : Γ × G → T and βj : G × Γ → T , weobtain the duality ω (cid:48) = β × βj : P (cid:48) × P → T . It is easy to see that ac-tually ω (cid:48) ( ξ, x ; x (cid:48) , ξ (cid:48) ) = ω ( x, ξ ; x (cid:48) , ξ (cid:48) ) , so we can recover the commutatorform as ω = ( β × βj ) ◦ ( j × P ) . Thus j describes how the symplecticstructure arises from the given duality β .Once the tilt map j : P → P (cid:48) is fixed, its extension to a homomor-phism J : H ( β ) → H ( β (cid:48) ) is unique provided J is required to leave T invariant: Using the fact that J should agree with j on G and Γ one immediataly obtains J (cid:0) c ( x, ξ ) (cid:1) = c (cid:104) ξ | x (cid:105) ( x, − ξ ) from the homo-morphism property. In the same way, the inverse tilt j ∗ : P (cid:48) → P with ( ξ, x ) (cid:55)→ ( − x, ξ ) can be extended to a unique homomorphism J ∗ : H ( β (cid:48) ) → H ( β ) , which is not the inverse of J : H ( β ) → H ( β (cid:48) ) .Under the above-mentioned identification H ( β (cid:48) ) ∼ = H ( β ) o , the homo-morphism J ∗ corresponds to ˇ J : H ( β ) o → H ( β ) , which is the sameas ˇ J : H ( β ) → H ( β ) o . Finally, the inversion map ( x, ξ ) (cid:55)→ ( − x, − ξ ) isa homomorphism on P since the latter is an abelian group. Similar to j N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 29 and j ∗ , it has a unique extension to a homomorphism ¯ J : H ( β ) → H ( β ) that leaves T invariant, unlike the inverion map on H ( β ) .Summing up, there are three natural (anti)homomorphism for chang-ing a left H ( β ) -module S into a left/right H ( β ) -module, which wedesignate by the same terminology as the maps themselves: • The module S ∧ := ˆ J ∗ ( S ) is called the forward twist of S . • The module S ∨ := ˇ J ∗ ( S ) is called the backward twist of S . • The module S − := ¯ J ∗ ( S ) is called the parity flip of S .If S is moreover a Heisenberg algebra , the torus acts naturally via ε T ,so the three derived modules above will also have the torus actingnaturally via ε T . Thus (left and right) Heisenberg algebras can beflipped as well as twisted forward and backward amongst themselves.It is well known [30, p. 95] that the category of P -modules over agroup P is isomorphic to the category of left modules over the groupring Z [ P ] , and this fact is much exploited in areas such as group homol-ogy [118, §6.1]. The same works for identifying K -modules (instead ofplain abelian groups) having a linear P -action with left modules overthe group algebra K [ P ] . We shall exploit a similar isomorphism forthe Heisenberg category ModH ( β ) , especially in the next subsectionfor the construction of various free objects. But in our case we have toaccount for one slight complication that requires Heisenberg modulesto be more than just modules over the group ring K [ H ( β )] , namely thenaturality of the torus action.A straightforward way to incorporate this requirement is a slight gen-eralization of K [ P ] , namely the so-called γ -twisted group algebra K γ [ P ] as described in [53, §3.6]. Given any cocycle γ ∈ Z ( P, K ∗ ) with K a trivial P -module, the definition of K γ [ P ] is the same as for K [ P ] as K -module, thus consisting of all maps P → K with finite support.But the multiplication is given by setting h z h z (cid:48) = γ ( z, z (cid:48) ) h zz (cid:48) for gen-erators h z , h z (cid:48) ( z, z (cid:48) ∈ P ) and extending by K -linearity. The cocyclecondition [53, (3.7)] for γ is equivalent to the associativity of K γ [ P ] .We shall now apply this construction to the phase space P = G ⊕ Γ of the Heisenberg group H ( β ) , endowed by the Heisenberg cocycle (cid:104) β (cid:105) ( x, ξ ; x (cid:48) , ξ (cid:48) ) := ε T (cid:104) ξ | x (cid:48) (cid:105) . One may view (cid:104) β (cid:105) : P × P → K as an extension of β : Γ × G → T via thenatural embeddings Γ , G (cid:44) → P and ε T : T → K . The cocycle conditionmay either be checked by a routine calculation or inferred from generalfacts about central group extensions [70, Cor. 5.2] since (cid:104) β (cid:105) is just the ε T -image of the factor set β ◦ ( π Γ × π G ) ∈ H ( P, T ) that describesHeisenberg extensions T (cid:26) H (cid:16) P by Lemma (cid:74) ?? (cid:75) . ∗ Definition 21.
Let H ( β ) = T G (cid:111) Γ be a Heisenberg group over aduality β . The Heisenberg group algebra is the twisted group alge-bra K (cid:104) β (cid:105) [ G ⊕ Γ] , which we denote by H K ( β ) .The Heisenberg group algebra is really almost the same as the plaingroup algebra, except that it works to identify the torus with the scalarring via ε T : ( T, · ) → ( K × , · ) . Lemma 22.
For any duality β , we have H K ( β ) ∼ = K [ H ( β )] / I T asan isomorphism of K -algebras, where the ideal I T is generated by all h c ( x,ξ ) − ε T ( c ) h x,ξ ) with c ( x, ξ ) ∈ H ( β ) .Proof. We define the evident group homomorphism ε : H ( β ) → H K ( β ) × by c ( x, ξ ) (cid:55)→ ε T ( c ) h x,ξ and extend it, via the universal property ofthe group algebra, to a K -algebra homorphism ε : K [ H ( β )] → H K ( β ) .Since ε is surjective, we obtain H K ( β ) ∼ = K [ H ( β )] / K T as K -algebras,with K T := ker( ε ) being an ideal of K [ H ( β )] . It is clear that I T ⊆ K T ,so it only remains to show the reverse inclusion.We introduce the T - degree of U = (cid:80) u ∈ H ( β ) λ u h u ∈ K [ H ( β )] as thenumber of nontrivial occurrences of c ∈ T for each u = c ( x, ξ ) involvedin U . More precisely, we set deg T ( U ) := { c ( x, ξ ) ∈ supp( U ) | c (cid:54) = 1 } , where supp( U ) denotes the support of U : H ( β ) → K , namely the setof those u ∈ H ( β ) for which λ u = U ( u ) (cid:54) = 0 . We prove K T ⊆ I T byinduction over T -degree.For the base case, let us take U ∈ K T with deg T ( U ) = 0 . In thatcase we have U = (cid:80) x,ξ λ x,ξ ) h x,ξ ) , hence ε ( U ) = (cid:80) x,ξ λ x,ξ ) h x,ξ = 0 and λ x,ξ ) = 0 for all ( x, ξ ) ∈ G ⊕ Γ . Now assume K T ⊆ I T forall U having T -degree below a fixed n > . Taking any U ∈ K T with deg T ( U ) = n we must show that U ∈ I T . By the T -degree hypothesis,we may write U = λ h c ( x,ξ ) + U (cid:48) with λ, c (cid:54) = 0 and deg T ( U (cid:48) ) < n .Denoting the generators of I T by i c,x,ξ := h c ( x,ξ ) − ε T ( c ) h x,ξ ) ∈ K T , itis clear that we have also U (cid:48)(cid:48) := U − λ i c,x,ξ = U (cid:48) + λ ε T ( c ) h x,ξ ) ∈ K T .Since deg T ( U (cid:48)(cid:48) ) < n , the induction hypothesis implies U (cid:48)(cid:48) ∈ I T andhence also U = U (cid:48)(cid:48) + λ i c,x,ξ ∈ I T ; this completes the induction step. (cid:3) As announced, the crucial fact is that we may identify Heisenbergmodules with modules over the twisted group algebra . (This does not generalize to Heisenberg algebras since in an algebra over H K ( β ) , anyelement h ,ξ ∈ H K ( β ) acts as scalar instead of an operator.) Lemma 23.
For any duality β , we have ModH ( β ) ∼ = H K ( β ) Mod asan isomorphism of categories.
N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 31
Proof.
We write Heisenberg modules as ( S, ∆ , η ) , where S is the un-derlying abelian group, the ring homomorphism ∆ : K → End Z ( S ) isthe structure map, and η : H ( β ) → Aut K ( S ) is the Heisenberg ac-tion. We have ∆ × ◦ ε T = η ◦ ι T by the definition of Heisenberg moduleswith ι T : T (cid:44) → H ( β ) the natural embedding. Similarly, H K ( β ) -modulescan be represented by ( S, ˜∆) , where ˜∆ : H K ( β ) → End( S ) is the cor-responding structure map. The isomorphism ModH ( β ) → H K ( β ) Mod sends a Heisenberg module ( S, ∆ , η ) to ( S, ˜∆) with ˜∆ (cid:16) (cid:88) ( x,ξ ) ∈ G ⊕ Γ λ x,ξ h x,ξ (cid:17) := (cid:88) ( x,ξ ) ∈ G ⊕ Γ ∆( λ x,ξ ) ◦ η (cid:0) x, ξ ) (cid:1) , where h x,ξ are the generators of H K ( β ) = K (cid:104) β (cid:105) [ G ⊕ Γ] . Its inversemaps the H K ( β ) -module ( S, ˜∆) to the Heisenberg module ( S, ∆ , η ) with structure map ∆ := ˜∆ ◦ ι K and Heisenberg action η := ˜∆ ◦ ε ,where ε : H ( β ) → H K ( β ) is defined by c ( x, ξ ) (cid:55)→ ε T ( c ) h x,ξ and where ι K : K (cid:44) → H K ( β ) is the embedding λ (cid:55)→ λ h , .Let us first check that the map Φ :
ModH ( β ) → H K ( β ) Mod iswell-defined. It is clear that ˜∆ is additive and even K -linear in thesense ˜∆( λ ˜ h ) = ∆( λ ) ◦ ˜∆(˜ h ) = ˜∆(˜ h ) ◦ ∆( λ ) for all λ ∈ K and ˜ h ∈ H K ( β ) . For ensuring that ˜∆ is a homomorphism of rings, it is thussufficient to check ˜∆( h x,ξ h x (cid:48) ,ξ (cid:48) ) = ˜∆( h x,ξ ) ˜∆( h x (cid:48) ,ξ (cid:48) ) . The left-handside comes out as ∆( ε T (cid:104) ξ | x (cid:48) (cid:105) ) ◦ η (cid:0) x + x (cid:48) , ξ + ξ (cid:48) ) (cid:1) by the defini-tion of the (cid:104) β (cid:105) -twisted multiplication in H K ( β ) . The right-hand sideis η (cid:0) (cid:104) ξ | x (cid:48) (cid:105) ( x + x (cid:48) , ξ + ξ (cid:48) ) (cid:1) = η ( ι T (cid:104) x (cid:48) | ξ (cid:105) ) ◦ η (cid:0) x + x (cid:48) , ξ + ξ (cid:48) ) (cid:1) , and thiscoincides with the left-hand side by ∆ ◦ ε T = η ◦ ι T . We conclude that Φ is well-defined.Next we consider the map Ψ : H K ( β ) Mod → ModH ( β ) . Since ι K isclearly a homomorphism of rings, the same is true of ˜∆ . One may alsocheck that ε and thus η is a group homomorphism. We have now that S is a K -module with an action η : H ( β ) → Aut K ( S ) , and it remains toshow ∆ × ◦ ε T = η ◦ ι T . But this follows from ι K ◦ ε T = ε ◦ ι T , which isevident. Thus Ψ is also well-defined.Finally, we prove that Φ ◦ Ψ = 1 HK ( β ) Mod and Ψ ◦ Φ = 1
ModH ( β ) .The former follows from ∆( λ ) ◦ η (cid:0) x, ξ ) (cid:1) = ˜∆( λ h , ) ◦ ˜∆( h x,ξ ) =˜∆( λ h x,ξ ) and the fact that ˜∆ is additive. The other identity is truebecause we have ˜∆( λ h , ) = ∆( λ ) ◦ η (cid:0) , (cid:1) = ∆( λ ) for all scalars λ ∈ K and also ˜∆( ε T ( c ) h x,ξ ) = ∆ (cid:0) ε T ( c ) (cid:1) ◦ η (cid:0) x, ξ ) (cid:1) = η (cid:0) c ( x, ξ ) (cid:1) for allHeisenberg operators c ( c, ξ ) ∈ H ( β ) , where the last equality uses ∆ ◦ ε T = η ◦ ι T . ∗ It is easy to check that this yields the desired pair of isomorphicfunctors, which actually leave the morphisms—viewed as set-theoreticmaps—invariant, since a map is H K ( β ) -linear precisely when it is aHeisenberg morphism. (cid:3) The reason why we have introduced the Heisenberg group algebra inthis subsection is that it exhibits the forward and backward twists aswell as the parity flip in a transparent manner. Indeed, it is easy tocheck that (9), (10) and (11) all stabilize the ideal I T of Lemma 22;thus one may pass to the quotient to obtain the following maps:Forward Twist ˆ J : H K ( β ) → H K ( β ) o , h x,ξ (cid:55)→ ε T (cid:104) ξ | x (cid:105) − h x, − ξ Backward Twist ˇ J : H K ( β ) → H K ( β ) o , h x,ξ (cid:55)→ ε T (cid:104) ξ | x (cid:105) − h − x,ξ Parity Flip ¯ J : H K ( β ) → H K ( β ) , h x,ξ (cid:55)→ h − x, − ξ Now ˆ J , ˇ J are involutions (involutive anti-automorphisms) of the K -algebra H K ( β ) while ¯ J = ˆ J ˇ J = ˇ J ˆ J is an involutive automorphism.Endowing K with the trivial involution, we see that (cid:0) H K ( β ) , ˆ J (cid:1) aswell as (cid:0) H K ( β ) , ˇ J (cid:1) is an involutive algebra (also known as a ∗ -algebra )over K . One may refer to (cid:0) H K ( β ) , ˆ J , ˇ J (cid:1) as a bi-involutive K -algebra.Let us conclude this subsection on the Heisenberg twist with theremark that, qua extensions, the Heisenberg groups H ( β ) and H ( β (cid:48) ) are the same: Indeed, using the notation from above, we have theequivalence(12) (cid:47) (cid:47) T (cid:31) (cid:127) (cid:47) (cid:47) T G (cid:111) Γ (cid:47) (cid:47) (cid:47) (cid:47) J (cid:15) (cid:15) G × Γ (cid:47) (cid:47) j (cid:15) (cid:15) (cid:47) (cid:47) T (cid:31) (cid:127) (cid:47) (cid:47) T Γ (cid:111) G (cid:47) (cid:47) (cid:47) (cid:47) Γ × G (cid:47) (cid:47) of central extensions, where the both projections are omission of thetorus component. (If one prefers to have identity for both marginalmorphisms, one may alter the second extension by using the projection T Γ (cid:111) G (cid:16) G × Γ with c ( ξ, x ) (cid:55)→ ( x, ξ ) instead.) But note that (12)is not a Heisenberg morphism in the sense of (1); see (cid:74) ?? (cid:75) . So, H ( β ) and H ( β (cid:48) ) are indeed distinct as Heisenberg groups , despite formingequivalent central extensions.2.4. Free Heisenberg Modules and Algebras.
It is always goodto have at one’s disposal various free objects, meaning left adjoints tovarious kinds of forgetful functor. Throughout this section, we fix acommutative and unital scalar ring K . N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 33
Let us start with the free Heisenberg module F β ( F ) over a set F and the free Heisenberg module F β ( M ) over a K -module M . In otherwords (overloading the same notation), we ask for the left adjoints F β of the forgetful functors ModH ( β ) → Set and
ModH ( β ) → K Mod .The latter functor
ModH ( β ) → H K ( β ) Mod is scalar restriction along ι : K (cid:44) → H K ( β ) , so its left adjoint is scalar extension along ι and wehave F β ( M ) ∼ = H K ( β ) ⊗ K M . Since adjoints respect composition offunctors [71, Thm. IV.8.1], we have F β ( F ) ∼ = F β ( K ( F ) ) with K ( F ) thefree K -module over the set F .For the sake of later reference, we state these free objects in explicitterms . Here and subsequently we shall not write out the action of thetorus T ≤ H ( β ) = T G (cid:111) Γ since it is always required to act via ε T ; weneed only the range ( x, ξ ) ∈ P = G ⊕ Γ ≤ H ( β ) . But note that here P is also written multiplicatively since we view it as a subgroup of H ( β ) . Proposition 24.
Let β be a duality, M a K -module and F a set.(1) We have F β ( M ) ∼ = (cid:77) ( y,η ) ∈ P yM η (cid:16) yM η ≡ { y } × M × { η } (cid:17) with ( x, ξ ) • yf η = ε T (cid:104) ξ | y (cid:105) ( xy ) f ξη for yf η ≡ ( y, f, η ) ∈ yM η .(2) We have F β ( F ) ∼ = K ( G × F × Γ) with the same action law but for K -basis elements yf η ≡ h y,f,η ∈ F β ( F ) .The embeddings ι : M → F β ( M ) and ι : F → F β ( F ) are a (cid:55)→ a .Proof. The notation yf η for elements in the free Heisenberg modulesis meant to convey the intuition of ground functions f modified by theaction of a scalar y and an operator η . Naturally, scalars are writtenmultiplicatively and operators additively.(1) Note that F β ( M ) is a K -module since each { y } × M × { η } ∼ = M is a distinct copy of the K -module M indexed by a particularphase point ( y, η ) ∈ P .For proving F β ( M ) ∼ = H K ( β ) ⊗ K M , consider the bilinear map H K ( β ) × M → F β ( M ) that sends (cid:80) ( y,η ) ∈ P λ y,η h y,η ∈ H K ( β ) and f ∈ M to (cid:80) ( y,η ) ∈ P λ y,η yf η . It descends to a K -linear map j : H K ( β ) ⊗ K M → F β ( M ) , which is clearly surjective. Forseeing that j is injective, assume (cid:80) ( y,η ) ∈ P λ y,η yf η = 0 . By thedefinition of the direct sum, λ y,η = 0 for all ( y, η ) ∈ P , and (cid:0) (cid:80) ( y,η ) ∈ P λ y,η h y,η (cid:1) ⊗ f = 0 . We conclude that j is a K -linearisomorphism. The canonical Heisenberg action of H K ( β ) ⊗ k M migrates to F β ( M ) via j , yielding the action law as stated. ∗ (2) Writing the generators by h f , we have K ( F ) ∼ = ⊕ f ∈ F [ h f ] , hencethe previous item implies F β ( F ) ∼ = (cid:77) ( y,η ) ∈ P (cid:77) f ∈ F { y } × [ h f ] × { η } , where the K -module on the right-hand side is free on the gen-erators ( y, h f , η ) . Mapping these to h y,f,η ∈ K ( G × F × Γ) = F β ( F ) yields the desired K -isomorphism and the corresponding actionlaw. (The notational convention is the same as before but re-stricted to the generators h f , which are shortened to f .)There is a simple alternative proof of item (2), which is not basedon item (1). Identifying Heisenberg module with modules over theHeisenberg group algebra, the free Heisenberg module over F is clearlygiven by H K ( β ) ( F ) . But as a K -module, we have H K ( β ) = K ( G × Γ) ,hence the K -isomorphism H K ( β ) ( F ) ∼ = K ( G × F × Γ) with h y,η h f ↔ h y,f,η .It is easy to see that the natural action of H K ( β ) ( F ) then induces theaction law stated. (cid:3) The action laws in Proposition 24 may be summarized by character-izing Heisenberg scalars x ∈ G as formal multipliers and Heisenbergoperators ξ ∈ Γ as formal exponents .Now we take the next step, constructing the free Heisenberg algebra P ( S ) over a given Heisenberg module S ∈ ModH ( β ) . Note that thisis not simply the free H K ( β ) -algebra over the H K ( β ) -module S , dueto the distinct roles of Heisenberg scalars and operators. Indeed, onemust take the symmetric algebra Sym KG ( S ) ≡ T KG ( S ) /I KG ( S ) with S considered as KG -module. By analogy with Propostion 24, we writethe given Heisenberg action of S as exponents. Proposition 25.
Given a Heisenberg module S ∈ ModH ( β ) , the in-duced free Heisenberg algebra is S (cid:44) → P ( S ) := Sym KG ( S ) , with action (13) ( x, ξ ) • s · · · s k = x ( s ξ ) · · · ( s ξk ) for s , . . . , s k ∈ S .Proof. Since a Heisenberg module S may be viewed as a module overthe Heisenberg group algebra H K ( β ) , the inclusion ι G : KG (cid:44) → H K ( β ) induces the structure of KG -module on S . Hence we may form thesymmetric algebra P ( S ) = Sym KG ( S ) , and it is easy to see that (13)yields a K -linear action on P ( S ) . By definition, the torus T ≤ H ( β ) acts naturally via ε T . Indeed, P ( S ) is a Heisenberg algebra since theaction of G ≤ H ( β ) coincides with the KG -scalar action on P ( S ) whileeach ξ ∈ Γ ≤ H ( β ) acts as K -algebra endomorphism. N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 35
For seeing that P ( S ) is free over S , we take a arbitrary Heisenbergalgebra U and Heisenberg morphism ϕ : S → U , where U is viewedas a Heisenberg module. We must show that ϕ extends to a unique AlgH ( β ) -morphism ˜ ϕ : P ( S ) → U . Being an algebra homomorphism,it must satisify ˜ ϕ ( s · · · s k ) = ϕ ( s ) · · · ϕ ( s k ) , which fixes ˜ ϕ uniquely.It remains to show that ˜ ϕ : P ( S ) → U defined in this way is actuallyan AlgH ( β ) -morphism. Using (H ) for the Heisenberg algebra U , it iseasy to see that ˜ ϕ is a KG -algebra homomorphism. For seeing that itrespects the action of Γ ≤ H ( β ) , one similarly applies (H ) for U . (cid:3) Note that the elements of the free Heisenberg algebra P ( S ) are es-sentially polynomials, hence the notation P . If S has generators ¯ S ,the symmetric algebra Sym KG ( S ) is generated by ¯ s · · · ¯ s k (¯ s i ∈ ¯ S ) as KG -module, and the action law may be expanded to ( x, ξ ) • x ¯ s · · · ¯ s k = ε T (cid:104) x | ξ (cid:105) x x (¯ s ξ ) · · · (¯ s ξk ) , which follows from (13) via (H ).We have proved that P : ModH ( β ) → AlgH ( β ) is left adjoint to theforgetful functor AlgH ( β ) → ModH ( β ) . We can combine the latterwith the left adjoints of Proposition 24 to obtain the free Heisenbergalgebra P β ( F ) over a set F and the free Heisenberg algebra P β ( M ) over a K -module M , which we describe now in more explicit terms. Corollary 26.
Let β be a duality, M a K -module and F a set.(1) We have P β ( M ) ∼ = Sym KG (cid:16) (cid:76) η ∈ Γ M η (cid:17) with action ( x, ξ ) • f η · · · f η k k = x f ξη · · · f ξη k k for ( f η , . . . , f η k k ) ∈ M η × · · · × M η k .(2) We have P β ( F ) ∼ = KG [ F × Γ] with action ( x, ξ ) • ( f η ) ν · · · ( f η k k ) ν k = x ( f ξη ) ν · · · ( f ξη k k ) ν k for f η i i ≡ ( f i , η i ) ∈ F × Γ ( i = 1 , . . . , k ) and ν , . . . , ν k ∈ N .The embeddings M → F β ( M ) and F → F β ( F ) are a (cid:55)→ a .Proof. Let β , M and F be as stated.(1) Writing M η := 1 M η and M Γ := (cid:76) η ∈ Γ M η ≤ F β ( M ) as abbre-viations, it is clear that M Γ generates F β ( M ) as KG -module,hence the f η · · · f η k k generate P β ( M ) as KG -module; the iso-morphism consists in multiplying out. The action law followsby combining that of F β ( M ) with (13).(2) The isomorphism Φ : P β ( F ) ∼ −→ KG [ F × Γ] is constructed bynoting that M := F β ( F ) = K ( G × F × Γ) ∼ = ( KG ) ( F × Γ) is a free KG -module with basis F × Γ so that P β ( F ) = S KG ( M ) is a ∗ polynomial ring over KG with indeterminates F × Γ . Again, theaction law follows by combining that of F β ( F ) with (13). (cid:3) The action law of Item 2 may be assimilated further to that of Item 1in Corollary 26 by using formal products η i ν i ∈ Γ × N in the exponents:In that case, Heisenberg operators act on their left components whileiterated multiplication acts on their right components.Up to now, we have now obtained the following five functors forgenerating the free slain and plain Heisenberg algebras: F β : Set → ModH ( β ) , F β : Mod K → ModH ( β ) , and P : ModH ( β ) → AlgH ( β ) , and P β = PF β : Set → AlgH ( β ) , P β = PF β : Mod K → AlgH ( β ) . On morphisms, these functors act in the usual way of free functors.For example, given any set map ζ : F → Φ , the induced morphism P β ( ζ ) : P β ( F ) → P β (Φ) in AlgH ( β ) is obtained by sending the KG -basis element ( f ξ ) ν · · · ( f ξ k k ) ν k ∈ P β ( F ) to ( ϕ ξ ) ν · · · ( ϕ ξ k k ) ν k ∈ P β (Φ) ,where ϕ := ζ ( f ) , . . . , ϕ k := ζ ( f k ) are the assigned image elements.The description of the corresponding free Heisenberg twain algebrasis somewhat more cumbersome. It is easier to describe first the generalsituation: Given an R -module M , we want to construct the free twainalgebra F over M , which is the smallest twain algebra ( F, + , (cid:63), · ) suchthat ( M, +) is a submodule of ( F, +) . We define a sequence of productalgebras ( C k × D k ) k> starting with C × D := M ⊕ M and construct-ing recursively C k +1 × D k +1 := Sym R ( D k ) (cid:63) × Sym R ( C k ) • , where thesubscripts on the symmetric algebras serve to keep the products apart.Using inducion, one checks immediately that C k ⊂ C k +1 ∧ D k ⊂ D k +1 ,so the ( C k × D k ) k> form an ascending sequence of R -algebras whosedirect limit we denote by(14) C ∞ × D ∞ := (cid:91) k> C k × D k . Obviously, ( C ∞ , (cid:63) ) and ( D ∞ , · ) are both algebras over R , and theircarriers coincide since C k ⊂ Sym R ( C k ) = D k +1 ⊂ D ∞ and conversely D k ⊂ C ∞ . Thus F := C ∞ = D ∞ is a twain algebra ( F, (cid:63), · ) over R ,and it is easy to see that it satisfies the required universal property forthe free twain algebra over the R -module M .The construction of the free Heisenberg twain algebra follows simi-lar lines, but the role of the ring R in the intertwined recursion stepsfor C k +1 × D k +1 is more subtle, reflecting the alternating scalar/operatorroles for the Heisenberg action on the recto/verso algebras. We shallmake use of the free functor P of Proposition 25, using an overbar N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 37 when referring to its verso variant (where G act as scalars and Γ asoperators). Proposition 27.
Given a Heisenberg module S ∈ ModH ( β ) , the freeHeisenberg twain algebra S (cid:44) → T ( S ) is defined as the direct limit (14) for the ascending sequence (15) C × D := S ⊕ S, C k +1 × D k +1 := P ( D k ) (cid:63) × ¯ P ( C k ) • with induced Heisenberg action.Proof. Let us first reassure ourselves that (15) is indeed an ascendingsequence: It is clear that each C k +1 = Sym KG ( D k ) is a recto plainHeisenberg algebra over β since Proposition 25 is applicable to D k viewed as a Heisenberg module. Similarly, each D k +1 = Sym K Γ ( C i ) isa verso plain Heisenberg algebra over β . It follows by joint inductionthat C k ≤ C k +1 in Alg KG and D k ≤ D k +1 in Alg K Γ . Moreover, theoperator actions for each are compatible, so the ( C k ) k> and ( D k ) k> are ascending sequences of (recto and verso) plain Heisenberg algebras.Hence we obtain in the direct limit (14) a recto plain Heisenberg alge-bra C ∞ and a verso plain Heisenberg algebra D ∞ .For making these into one twain Heisenberg algebra, we use the over-lay C ∞ (cid:62) ι D ∞ introduced before Example 20, with the transfer isomor-phism ι : C ∞ ∼ −→ D ∞ defined as follows. We keep C = M = D invari-ant. Then let s ∈ C ∞ be an element of rank k > so that s is containedin C k but in no earlier stage. We can write such elements as K -linearcombinations of s = [ s ] (cid:63) · · · (cid:63) [ s m ] for s , . . . , s m ∈ D k − , with [ − ] denoting the generator embedding. We set ι ( s ) = [ s ] ∈ D k +1 ≤ D ∞ if m > and ι ( s ) = s ∈ D k − ≤ D ∞ if m = 1 . It is easy to see that ι is bijective with ι − having an analogous description. The resultingproduct works in the expected manner, for example ([ s ] (cid:63) [ s ]) · ([ t ] (cid:63) [ t ]) = [[[ s ] (cid:63) [ s ]] · [[ t ] (cid:63) [ t ]]] , expressed without embeddings by the apparently vacuous statement(similar to the corresponding statement for multiplying polynomials):The pointwise product of s (cid:63) s and t (cid:63) t yields ( s (cid:63) s ) · ( t (cid:63) t ) .For checking that T ( S ) is the free Heisenberg twain algebra over S ,let ϕ : S → T be a Heisenberg morphism to an arbitrary Heisenbergtwain algebra T . We must determine a unique map ˜ ϕ : T ( S ) → T that factors throught the embedding S (cid:44) → T ( S ) . Obviously, we mustset ˜ ϕ ( s ) = ϕ ( s ) for all s ∈ S , and this in fact determines ˜ ϕ on all of T ( S ) This presupposes that embedded elements [ s ] ∈ C k embed into D k +1 as theiroriginal, thus identifying [[ s ]] = s ∈ D k − ; the same is assumded for embeddingsinto C k +1 . We have implicitly used this for establishing C k ≤ C k +1 and D k ≤ D k +1 . ∗ since it must be a homomorphism (with respect to both multiplicationmaps). In effect, one recursively replaces brackets by ˜ ϕ until hittingon elements of S . We have thus established uniqueness, and we knowthat ˜ ϕ is a twain homomorphism. Finally, one checks that ˜ ϕ respectsthe Heisenberg action using induction on rank: While the action of cx ∈ T G involves tracking down one path to a single leaf s ∈ S , the actionof ξ ∈ Γ precipitates down to all the leaves. (cid:3) Proof.
Let us also sketch an alternative proof via universal algebra . Tothis end, consider the signature [ AlgH ( β )] defined over the signatureof abelian groups (binary plus, unary minus, nullary zero), togetherwith the two binary products (cid:63) and · , the unary homotheties u · forall Heisenberg actors u ∈ H ( β ) , and the unary homotheties λ · for thescalars λ ∈ K . The signature [ AlgH ( β ) S ] is obtained by adding asconstants (nullary operations) all elements of S to capture the em-bedding of S . The variety of Heisenberg twain algebras AlgH ( β ) isdefined over the signature [ AlgH ( β )] subject to the K -algebra axioms(distributivity doubled for (cid:63) and · ), the laws (H )–(H ), and two copies(one for each product) of (H ) and (H ). We add to this the Heisenbergaction for each element of S , to obtain the variety AlgH ( β ) S over thesignature [ AlgH ( β ) S ] .Note that we do not have generators since the embedding of S servesthis purpose. Our task now is to establish an AlgH ( β ) S isomorphism i from the term algebra T of the variety AlgH ( β ) S to the free Heisen-berg twain algebra T ( S ) as defined in the previous proof. We do this byorienting the laws of AlgH ( β ) S in such a way that the correspondingnormal forms can be identified with elements of T ( S ) .Representing by the empty sum, it is clear that distributivity overthe products (cid:63) and · as well as the homotheties of H ( β ) and K maybe oriented in the usual manner to reduce each element in T to asum of terms (meaning elements of T that do not contain + , − , ).Moreover, the linearity of the Heisenberg action allows us to writethe K -homotheties on the very front, so everything is reduced to K -linear combinations of monomials (terms that do not contain scalarsfrom K ). Using (H ), we can furthermore eliminate Heisenberg actorsinvolving c ∈ T . Applying both copies of (H ) and (H ) allows us tomove the remaining Heisenberg actors x ∈ G and ξ ∈ Γ all the wayto the embedded elements s ∈ S , where we can apply the importedHeisenberg action.Thus it remains to show how to identify Heisenberg-free monomials(elements of T involving only (cid:63) and · as well as the embedded ele-ments s ∈ S ). But it should be clear how to do this from the informal N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 39 example mentioned in our earlier proof above. It is also straightfor-ward to check that the resulting map i is indeed an isomorphism in thevariety AlgH ( β ) S . (cid:3) The reader will agree that the above construction if full of tedioustechnicalities but utterly simple from the structural perspective: Asusual, the elements of the free Heisenberg twain algebra are built upby schematically applying the available operations.3.
Fourier Operators in Algebra
The Notion of Fourier Doublets.
Heisenberg twists ˆ J : H ( β ) → H ( β ) o and ˇ J : H ( β ) → H ( β ) o and the parity flip ¯ J : H ( β ) → H ( β ) in §2.3. They induce functors S (cid:55)→ S ∧ and S (cid:55)→ S ∨ from the cat-egory of left/right to the category of right/left Heisenberg algebras,and an endofunctor S (cid:55)→ S − on the category of left/right Heisen-berg algebras. Note that while the latter functor may be interpretedas AlgH ( − Γ × − G , T ) with − Γ : Γ → Γ and − G : G → G being thenegation maps, this does not work for the former functors since thetwists are not induced by morphisms of Du .We will introduce forward/backward Fourier operators as Heisenbergmorphisms over the forward/backward twists and reversal operators asHeisenberg morphisms over the parity flip. We can express this via thecorresponding modules as follows (reversal is only mentioned for rightmodules Σ but is defined in the same way for left modules S ). Definition 28.
For a fixed duality β , let S be a left and Σ a rightHeisenberg algebra (slain or plain or twain). • A Heisenberg morphism (cid:70) ∧ : S → Σ ∧ is called a forward Fourieroperator from S to Σ , • and a Heisenberg morphism (cid:70) ∨ : S → Σ ∨ a backward Fourieroperator from S to Σ , • a Heisenberg morphism (cid:80) : Σ → Σ − a reversal operator on Σ .By default, the term Fourier operator refers to the forward kind.Since the twists concur to cycles of periodicity four, their action maybe visualized on the
Heisenberg clock of Figure 2. One may think of thisas a period-four (“complexified”) analog of duality in finite-dimensionalvector spaces. Topologically more accurate, we may put a Möbius striparound a clock face (left half ⤹ standing for left modules, right half ⤸ for right modules), moving in six-hour steps: One starts with S at h to reach S ∧ at h , moving on to S − opposite of S at h , stoppingat S ∨ opposite of S ∧ at h , finally returning to S next day at h . ∗ Original ΣΣ ∧ Forward Twist Σ - Parity Flip Backward Twist Σ ∨ J * ⋀ J * ⋁ J * ⋁ J * ⋀ S ℱ ∧ ℱ ∨ Figure 2.
The Heisenberg Clock
Remark 29.
Though the complexified duality structure seems to be anunnecessary complication , it will be crucial for understanding Fourierinversion (§3.3). Indeed, a single (forward) Fourier operator may becharacterized in a much simpler way (taking all modules left):A conjugate-linear map between modules M and M (cid:48) over an involu-tive algebra A is a group homomorphisms ϕ : ( M, +) → ( M (cid:48) , +) suchthat ϕ ( a · m ) = a ∗ ϕ ( m ) for all a ∈ A and m ∈ M , where a (cid:55)→ a ∗ de-notes the involution. Viewing Heisenberg modules S ∈ ModH ( β ) asmodules over the involutive algebra (cid:0) H K ( β ) , ˆ J (cid:1) , the Fourier operatorsfrom S to Σ are precisely the conjugate-linear maps. (Of course, oneobtains backward Fourier operators by taking the backward twist ˇ J instead of the forward twist ˆ J .)One might be tempted to simplify matters even more by taking thecodomain to be just S (cid:48) := Σ ∧ ; then a (forward) Fourier operator isan H K ( β ) -linear map S → S (cid:48) simpliciter . While this may be doneon an adhoc basis for isolated examples, it does not mesh smoothlywhen considering forward and backward Fourier operators in tandem:The inverse operator is necessarily defined on a module with modifiedHeisenberg action (see Definition 52). Moroever, in the all-importantPontryagin setting (§3.2), the definition of the actions (22)–(23) wouldincur spurious signs concealing the twists. (cid:125) We will sometimes write ˆ s := (cid:70) ∧ ( s ) ∈ Σ ∧ and ˇ s := (cid:70) ∨ ( s ) ∈ Σ ∨ for the forward/backward Fourier transform of s ∈ S . While the hat N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 41 notation ˆ s for the (forward) Fourier transform is very commonly usedin engineering practice, one may also find the check notation ˇ s for thebackward transform in some places like [45, Def. 31.16], where it isspecifically introduced in the L setting (confer Proposition 54 below).For the sake of uniformity, we write also ¯ s := (cid:80) ( s ) and ¯ σ := (cid:80) ( σ ) forthe reversal of s ∈ S and σ ∈ Σ .Writing ˆ h = ˆ J ( h ) , ˇ h = ˇ J ( h ) for the twists and ¯ h = ¯ J ( h ) for the flips of Heisenberg actors h ∈ H ( β ) , we can characterize forward/backwardFourier operators and reversal operators as follows:(16) (cid:70) ∧ ( h • s ) = ˆ s • ˆ h, (cid:80) ( h • s ) = ¯ h • ¯ s (cid:70) ∨ ( h • s ) = ˇ s • ˇ h, Using the hat/check notation on Heisenberg actors, the basic cycle inthe Heisenberg clock (Figure 2) reads h (cid:55)→ ˆ h (cid:55)→ h − (cid:55)→ ˇ h (cid:55)→ h . To avoidconfusion with honest Heisenberg morphisms, we use cycle markers in (cid:70) ∧ : S (cid:55)→ Σ , (cid:80) : S (cid:55) (cid:55)→ S, (cid:70) ∨ : S (cid:55) (cid:55) (cid:55)→ Σ for expressing the morphisms of Definition 28. In fact, we will mostlyneed the (cid:55)→ notation in the sequel.In terms of the bimodule structure mentioned after (H )–(H ) above,the conditions (16) decompose into the requirement of respecting thetorus action together with (cid:70) ∧ ( x • s ) = (cid:70) ∧ ( s ) • x, (cid:70) ∧ ( ξ • s ) = (cid:70) ∧ ( s ) • ξ − , (17) (cid:80) ( x • s ) = x − • (cid:80) ( s ) , (cid:80) ( ξ • s ) = ξ − • (cid:80) ( s ) , (18) (cid:70) ∨ ( x • s ) = (cid:70) ∨ ( s ) • x − , (cid:70) ∨ ( ξ • s ) = (cid:70) ∨ ( s ) • ξ (19)for ( x, ξ ) ∈ G × Γ and s ∈ S . Here x (cid:55)→ x − and ξ (cid:55)→ ξ − denote thenegation maps of G and Γ , respectively.The parallel treatment of forward and backward Fourier operators,while appealing from an aesthetic viewpoint, is not economic for al-gorithmic purposes. In the classical scenario described in §3.2, thedistinction between (cid:70) ∧ and (cid:70) ∨ hinges on the sign in the exponential.One can generate one from the other by applying a sign change, whichis incorporated in a distinguished reversal operator . Definition 30.
Let β be a duality. We call S ∈ AlgH ( β ) symmetric if it is endowed with an involutive reversal operator (cid:80) : S → S .In this case (cid:70)(cid:80) is a backward/forward Fourier operator iff (cid:70) is aforward/backward Fourier operator. It is then preferrable to distin-guish, say, some forward Fourier operator (cid:70) and retain (cid:70) ∨ := (cid:70)(cid:80) asan abbreviation for the derived backward Fourier operator . For empha-sizing the underlying symmetry, one may still employ (cid:70) ∧ := (cid:70) as anotational variant for the given Fourier operator. ∗ A Fourier operator (cid:70) from a symmetric Heisenberg algebra S toanother symmetric Heisenberg algebra Σ is called symmetric if it com-mutes with the reversal operators in the sense that (cid:70)(cid:80) S = (cid:80) Σ (cid:70) . Weshall henceforth suppress the domain of the reversal operators, writingagain (cid:80) when no confusion arises. Moreover, we assume all Heisenbergalgebras and Fourier operators as symmetric (see Definitions 32 below),because all natural examples appear to be like this.Symmetric Heisenberg algebras also suggest the following convenientjargon. We call an operator (cid:71) : Σ → S sign inverse to (cid:70) if (cid:71)(cid:70) = (cid:80) S and (cid:70)(cid:71) = (cid:80) Σ . In terms of forward/backward Fourier operators: Theinverse of (cid:70) ∧ is the sign inverse of (cid:70) ∨ , and the inverse of (cid:70) ∨ the signinverse of (cid:70) ∧ . Remark 31.
Before we now introduce the central object of our alge-braic approach to Fourier analysis, let us make a brief comparison withdifferential algebra [60, 89]. Faithful to its name as a discipline, its cen-tral algebraic objects are differential algebras, viz. algebras with distin-guished derivations. Likewise, in our case we will introduce
Heisenbergalgebras with distinguished
Fourier operators . There are, however, twonoteworthy differences:(1) In general we cannot expect Fourier operators to have the samedomain and codomain , even if the signal and spectral spacescoincide: In the presence of an algebra structure, Fourier op-erators are only K -linear endomorphisms but not as algebraendomorphisms (see Definition 52).(2) While one typically has a great variety of derivations on anygiven ring (they form a Lie algebra!), the variety of Fourieroperators between fixed Heisenberg algebras appears to be ratherrestricted, at least under the usual topological constraints.In typical cases (see Remark 38), the Heisenberg structure of the do-main spawns the Fourier operator with its codomain—but this is some-thing that the algebra “does not see”. (cid:125)
With these qualifications in mind, we can now proceed to definingan appropriate algebraic notion of Fourier structures. Since it harborsa pair of Heisenberg algebras, we will call such an object a
Fourier dou-blet . As we shall see in the sequel, a Fourier doublet may occasionallycoalesce into a Fourier singlet (Definition 52). Considering the greatrole of spectroscopy as an early motivation for classical Fourier analy-sis, the doublet/singlet metaphor does not seem to be out of place (seealso Remark 33).
N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 43
Definition 32.
Let β be a duality. Then ( S, Σ , (cid:70) ) is a Fourier doublet over β if (cid:70) : S (cid:55)→ Σ is a Fourier operator between the left Heisenbergalgebra S ∈ AlgH ( β ) and the right Heisenberg algebra Σ ∈ AlgH ( β ) .Going back to the definition of (plain) Heisenberg algebras, it willbe seen that the essential property required in a Fourier doublet isthe so-called convolution theorem (cid:70) ( s (cid:63) s (cid:48) ) = (cid:70) s · (cid:70) s (cid:48) . The choice ofaxioms for the axiomatization in this paper is vindicated by the resultsderived in [63, Thm. 2.1]. It is shown there that Fourier operatorsare essentially characterized uniquely by the (forward and backward)convolution theorems, at least in the important case of the Schwartz-Bruhat functions to be treated below (Theorem 58).In the sequel, Fourier doublets will be written (cid:68) = [ (cid:70) : S (cid:55)→ Σ] .When referring to elements of the doublet (cid:68) , we identify the latterwith their graphs. In other words, every d ∈ (cid:68) is a pair d = ( s, σ ) ∈ S × Σ such that σ = (cid:70) s . Following classical usage [24, §2; Prob. 6.30],we call d a Fourier pair . Unlike Bracewell (who writes them s ⊂ σ ), we prefer the suggestive notation d = [ s (cid:55)→ σ ] for Fourier pairsin (cid:68) = [ (cid:70) : S (cid:55)→ Σ] . Building on widespread conventions in signaltheory [16, 24], we refer to the s ∈ S as signals and to the σ ∈ Σ as spectra . Accordingly, we call S the signal space and Σ the spectralspace of the Fourier doublet (cid:68) . Remark 33.
While the “spectrum of a signal” has an immediate physi-cal interpretation (in optics and acoustics), there is also a deep relationbetween Fourier analysis in the Pontryagin setting (Theorem 37) and classical spectral theory : As detailed in Proposition 1.15 and Theo-rem 1.30 of [37], the Fourier transformation (cid:70) : L ( G ) → C (Γ) ex-tends canonically to the group C ∗ -algebra (cid:65) = C ∗ ( G ) and its unital-ization, yielding the Gelfand transformation (cid:65) → C (cid:0) σ ( (cid:65) ) (cid:1) , a (cid:55)→ ˆ a .Here σ ( (cid:65) ) is the algebra spectrum of (cid:65) , meaning its maximal idealspace, and ˆ a the homeomorphism from σ ( (cid:65) ) to the classical operatorspectrum σ ( a ) = { λ ∈ C | λ (cid:65) − a is singular } . (cid:125) The category of Fourier doublets over a duality β , denoted by Fou ( β ) ,is defined as the full subcategory of the arrow category AlgH ( ) → gen-erated by Fourier doublets over β ; the corresponding morphisms arecalled Fourier morphisms over β . In detail, given two Fourier doublets (cid:68) = [ (cid:70) : S (cid:55)→ Σ] and (cid:68) (cid:48) = [ (cid:70) (cid:48) : S (cid:48) (cid:55)→ Σ (cid:48) ] , a Fourier morphism from (cid:68) to (cid:68) (cid:48) has the form ( a, α ) with a left Heisenberg morphism a : S → S (cid:48) and a right Heisenberg morphism α : Σ → Σ (cid:48) such that α (cid:70) = (cid:70) (cid:48) a . Werefer to a and α , respectively, as the signal map and spectral map of ∗ the Fourier morphism. Following a similar procedure as for the cate-gory AlgH ( ) , we have the fibration(20)
Fou ( ) = (cid:93) β ∈ Du Fou ( β ) making up the category of all Fourier doublets. Note that the Heisen-berg algebras S and Σ in a Fourier doublet [ (cid:70) : S (cid:55)→ Σ] may be slain,plain or twain—depending on how many nontrivial multiplications theycome with. Naturally, we shall also write Fou ( T ) for the full sub-category of Fou ( ) obtained by restricting the disjoint union in (20)to β ∈ Du ( T ) .As noted in §2.2, the category AlgH ( β ) has products. It is theneasy to see that the same is true of Fou ( β ) . Indeed, given doublets (cid:68) and (cid:68) (cid:48) as above, it is easy to see that (cid:70) × (cid:70) (cid:48) : S × S (cid:48) → Σ × Σ (cid:48) isa Fourier operator so that the product doublet (cid:68) × (cid:68) (cid:48) is the Fourierdoublet [ (cid:70) × (cid:70) (cid:48) : S × S (cid:48) (cid:55)→ Σ × Σ (cid:48) ] .In §2.4 we have constructed the free Heisenberg module/algebra. Weshall now package them to create free Fourier doublets . To this end, weuse the following basic result in category theory whose proof is routine. Lemma 34.
Let (cid:67) be a concrete category with free functor Z : Set → (cid:67) . Then the functor Z → : Set → → (cid:67) → with Z → ( X x → X ) := (cid:16) Z ( X ) Z ( x ) −→ Z ( X ) (cid:17) is free, being left adjoint to the forgetful functor U → : (cid:67) → → Set → thatsends C c → C to the set map U ( C ) U ( c ) −→ U ( C ) . With this lemma, one can establish the free doublets generated byan arrow in
Set or in
Mod K . Proposition 35.
Let β be a duality and f : L → Λ a set map or K -module homomorphism.(1) Setting ˜ f := F β ( f ) , the free slain doublet over f : L → Λ is [ ˜ f : F β ( L ) (cid:55)→ F β (Λ) ∧ ] .(2) Setting ˜ f := P β ( f ) , the free plain doublet over f : L → Λ is [ ˜ f : P β ( L ) (cid:55)→ P β (Λ) ∧ ] .(3) Setting ˜ f := T β ( f ) , the free twain doublet over f : L → Λ is [ ˜ f : T β ( L ) (cid:55)→ T β (Λ) ∧ ] . Classical Pontryagin Duality.
Now is a good time to contem-plate the most crucial example of a Fourier doublet in classical Fourieranalysis—the
Fourier transform on LCA groups. This is in fact a very
N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 45 extensive class of examples since one may start from an arbitrary Pon-tryagin duality.Recall from Example 5 that for any LCA group G there is an LCAgroup Γ = ˆ G , called the group dual to G , such that the natural pairing(21) (cid:36) G : G × Γ → T , ( x, ξ ) (cid:55)→ (cid:104) x | ξ (cid:105) ≡ ξ ( x ) is a duality, known as the Pontryagin duality for G and Γ . As long asno confusion is likely, we shall suppress the index and just write (cid:36) forthe Pontryagin duality in question.Both LCA groups G and Γ give rise to natural algebras, which areconnected by the Fourier transform. In detail, we have the complex vec-tor space L ( G ) consisting of all functions on the topological group G that are absolutely integrable with respect to Haar measure. It is wellknown that they form a complex algebra under convolution (cid:63) , whichwe call the convolution algebra . Some sources [100, p. vi] dub it the(topological) group algebra of (cid:36) .We define a left Heisenberg action H ( (cid:36) ) × L ( G ) → L ( G ) by lettingthe torus T (cid:44) → C × act naturally via the embedding while setting ( x • s )( y ) = s ( y − x ) , ( ξ • s )( y ) = (cid:104) ξ | y (cid:105) s ( y ) (22)for all x ∈ G , ξ ∈ Γ and s ∈ L ( G ) . Here G • L ( G ) ⊆ L ( G ) followsfrom translation invariance of Haar measure while Γ • L ( G ) ⊆ L ( G ) is clear because |(cid:104) ξ | y (cid:105)| ≤ . We refer to the two actions of (22), re-spectively, as translation and modulation on G because of their mostimportant instantiation (Example 40a).On the dual group Γ , we set up the space C (Γ) of bounded continu-ous functions Γ → C vanishing at infinity as in [100, A11]. Clearly, thisis a complex algebra (cid:0) C (Γ) , · (cid:1) under pointwise multiplication, whichwe call the pointwise algebra of (cid:36) . Setting up the right Heisenbergaction C (Γ) × H ( (cid:36) ) → C (Γ) in complete analogy to the convolutionalgebra, we define translation and modulation on Γ by ( σ • ξ )( η ) = σ ( η − ξ ) , ( σ • x )( η ) = (cid:104) η | x (cid:105) σ ( η ) . (23)The closure properties are again evident: We have Γ • C (Γ) ⊆ C (Γ) by the continuity of η (cid:55)→ ξ + η and G • C (Γ) ⊆ C (Γ) by that of (cid:104) x |(cid:105) .The torus action is of course again via the embedding T (cid:44) → C × . Clearly, this reduces to the plain group algebra [62, §II.3] when G is consideredfrom a purely algebraic viewpoint, i.e. given the discrete topology). ∗ With the algebras L ( G ) and C (Γ) in place, we can now define the forward and backward Fourier transform (24) (cid:40) (cid:70) ∧ : L ( G ) (cid:55)→ C (Γ) , (cid:70) s ( ξ ) := (cid:114) G (cid:104) ξ | + y (cid:105) s ( y ) dy, (cid:70) ∨ : L ( G ) (cid:55) (cid:55) (cid:55)→ C (Γ) , (cid:70) s ( ξ ) := (cid:114) G (cid:104) ξ |− y (cid:105) s ( y ) dy. Of course, we have to ensure that they are indeed Fourier operatorsand thus deserve their name (Theorem 37 below).
Remark 36.
Fourier transforms are plagued with a multitude of ar-bitrary conventions (overall factors, signs and factors in the exponent,signs in the Heisenberg actions, etc.), and there seems to be no com-pelling a priori reason as to which transform in (24) should be chosenforward and which backward. We may refer to the two possibilitiesas the forward-positive and the forward-negative sign conventions. Us-ing this jargon, we have thus adopted the forward-positive conventionin this paper. In the Chapter “A Plus or Minus Sign in the FourierTransform?” of the applied monograph [114] on electron holography,the authors have made the same choice:The sign of the exponential in the Fourier transform issomething that we have been concerned with for manyyears. Of course, there are two conventions that havebeen used with almost equal frequency [...] we have usedthe convention of the positive sign in the exponential forthe forward transform which represents the Fraunhoferdiffraction pattern for a real-space object [...] If the otherconvention is to be used for the Fourier transform expo-nent sign, then all authors should be advised of all theseother implications, which are not immediately obvious.Otherwise, we might find ourselves producing a treat-ment of positron holography!While there may be physical reasons for preferring one or the otherconvention, there is little ground for preference outside applications.In classical Fourier analysis (Example 40a), both sign conventions areto be found—see for example [24] versus [108], and note the
Warning on page 29 of [108]. In abstract Fourier analysis , however, the forward-minus convention appears to be more common [37], [100], [67].We have picked the forward-plus convention (24) since it meshesnicely with our abstract approach (see the Heisenberg clock in Fig-ure 2): After fixing the Heisenberg group in the form H ( β ) = T G (cid:111) Γ ,the tilt map j : P → P (cid:48) is the natural choice to march “forward” (ex-ample (b) in §2.3), which induces the forward twist in the form (9). N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 47
But of course this does not mean our setup is written in stone: In dif-ferent circumstances, other combinations of the various conventions—Heisenberg group, Heisenberg action, Heisenberg twists—may prove tobe better suited. (cid:125)
As explained after Definition 30, we can also define (cid:70) ∨ from (cid:70) := (cid:70) ∨ since we have the natural reversal operators (cid:80) : L ( G ) (cid:55) (cid:55)→ L ( G ) as wellas (cid:80) : C (Γ) (cid:55) (cid:55)→ C (Γ) with ( (cid:80) s )( y ) = s ( − y ) and ( (cid:80) σ )( η ) = σ ( − η ) .Together with these, the convolution algebra L ( G ) and the pointwisealgebra C (Γ) make up the prototypical example of a Fourier doublet . Theorem 37.
Let G and Γ be LCA groups under Pontryagin duality (cid:36) : Γ × G → T . Then [ (cid:70) : L ( G ) (cid:55)→ C (Γ)] is a Fourier doublet withreversal operators (cid:80) : L ( G ) (cid:55) (cid:55)→ L ( G ) and (cid:80) : C (Γ) (cid:55) (cid:55)→ C (Γ) .Proof. All these facts are very simple or otherwise well-known, so forthe most part it will suffice to provide suitable pointers to the literature.We have to check the following facts:(1)
Convolution algebra:
We refer to Theorems 1.1.6/1.1.7 in [100]for the well-known fact that (cid:0) L ( G ) , (cid:63) (cid:1) is a Banach algebra.Since (22) are group actions, the laws (H )–(H ) are satisfied,and it suffices to verify conditions (H )–(H ). The relation (H )between convolution and translation is well-known [37, p. 51].For checking (H ), we evaluate the right-hand side ( s • ξ ) (cid:63) (˜ s • ξ ) at a point y ∈ G to obtain (cid:114) G (cid:104) ξ | z − y (cid:105) s ( y − z ) (cid:104) ξ | − z (cid:105) ˜ s ( z ) dz = (cid:104) ξ | − y (cid:105) (cid:114) G s ( y − z ) ˜ s ( z ) dy = (cid:104) ξ | − y (cid:105) ( s (cid:63) ˜ s )( y ) , which is the left-hand side ( s (cid:63) ˜ s ) • ξ evaluated at y . The torusaction law (H ) holds trivially for the embedding T (cid:44) → C × .Finally, (H ) follows from the fact that (cid:104) ξ |(cid:105) is a homomorphism.We have thus verified that L ( G ) ∈ AlgH ( (cid:36) ) .(2) Pointwise algebra:
Again, it is well known that (cid:0) C (Γ) , · (cid:1) is aBanach algebra; see for example Appendix A12 in [100]. It isagain clear that (23) constitute group actions, so it suffices tocheck (H )–(H ). This time, (H ) follows directly from the asso-ciativity of ( C , · ) while (H ) is the statement that translation isa homomorphism and (H ) is again trivial. It remains to showthe transposed version of (H ), namely ( σ • ξ ) • x = (cid:104) ξ | x (cid:105) ( σ • x ) • ξ ,which now follows from (cid:104)| x (cid:105) being a homomorphism. This es-tablishes C (Γ) ∈ AlgH ( (cid:36) (cid:48) ) .(3) Fourier transform:
It is well-known that the Fourier transform (cid:70) = (cid:70) ∧ of (24) is a homomorphism (cid:0) L ( G ) , (cid:63) (cid:1) → (cid:0) C (Γ) , · (cid:1) ∗ of C -algebras; see for example Theorems 1.2.2 and 1.2.4(b)of [100]. Hence (cid:70) respects also the trivial torus action, andit remains to show the two relations (17). They follow fromthe homomorphism property, respectively, of (cid:104) η |(cid:105) and (cid:104)| y (cid:105) , em-ploying a linear substitution in the integral and appealing totranslation invariance of Haar measure on G . The correspond-ing relations (19) for (cid:70) ∨ are automatic since (cid:70) is symmetric(Item 4 below), but they can be established directly in an anal-ogous manner.(4) Reversal operators:
It is easy to see that (cid:80) : L ( G ) → L ( G ) and (cid:80) : C (Γ) → C (Γ) are involutive reversal operators, soboth L ( G ) and C (Γ) are symmetric Heisenberg algebras. Forseeing that (cid:70) commutes with the reversal operators, one usesthe substitution y (cid:55)→ − y in the integral.It is easy to see that pre- or postcomposing by the reversal operators (cid:80) exchanges (cid:70) ∧ and (cid:70) ∨ . (cid:3) Remark 38.
Using the operator e of evaluation at ∈ G , just as inExample 1 of [99] where G = R , the definition of the Fourier trans-formation (24) may be written in the concise form (cid:70) ∧ s ( ξ ) = e ( s (cid:63) (cid:80) ξ ) and (cid:70) ∨ s ( ξ ) = e ( s (cid:63) ξ ) . But note the following provisos: In general, thecharacters ξ ∈ Γ are not elements in L ( G ) ; they are only when G iscompact. Nevertheless, they are always in L ∞ ( G ) so that the convolu-tion s (cid:63) ξ is continuous by Proposition (2.39d) of [37]. This is why onemay apply the evaluation operator e , which is not normally possible onfunctions in L ( G ) . In fact, we follow here the purely algebraic setting as in differential algebra (where functions are viewed as elements in aring carrying a derivation), so evaluation is not available: neither forthe continuous elements of the signal space S = L ( G ) nor for those ofthe spectral space Σ = C (Γ) , so also the left-hand side (cid:70) s ( ξ ) of theabove definition is not feasible in our present setting. (cid:125) Remark 39.
It should also be mentioned that Fourier operators aresometimes used like quantifiers . So if T is a term containing the freevariable x such that x (cid:55)→ T constitutes a signal in L ( G ) , we shallwrite (cid:70) x T for the spectrum (cid:70) ( x (cid:55)→ T ) . This may be somewhat pedan-tic (in practical applications the subscript x is often suppressed), butit may prevent ambiguities. (cid:125) We refer to [ (cid:70) : L ( G ) (cid:55)→ C (Γ)] as the classical Fourier doublet ofthe Pontryagin dualty (cid:36) : Γ × G → T . As a shorthand, we shall alsowrite this doublet as L (cid:104) Γ | G (cid:105) (cid:36) or briefly L (cid:104) Γ | G (cid:105) when the Pontryaginduality is clear from the context. It is easy to see that (cid:36) (cid:55)→ L (cid:104) Γ | G (cid:105) (cid:36) N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 49 is a functor Du ( T ) → Fou ( T ) . We transfer the symmetric monoidalstructure of Du ( T ) to its essential image (generally defined as the fullsubcategory generated by the image objects). Up to isomorphism, wethus define the tensor product doublet L (cid:104) Γ | G (cid:105) (cid:36) ⊗ L (cid:104) Γ (cid:48) | G (cid:48) (cid:105) (cid:36) (cid:48) := L (cid:104) Γ ⊕ Γ (cid:48) | G ⊕ G (cid:48) (cid:105) (cid:36) ⊗ (cid:36) (cid:48) , with Fourier operator (cid:70) ⊗ (cid:70) (cid:48) : L ( G × G (cid:48) ) (cid:55)→ C (Γ × Γ (cid:48) ) . Obviously,we may extend this to tensor products with finitely many factors.Using Fubini’s theorem [78, Prop. I.46], it is easy to see that Fourieroperators are multiplicative . To make this precise, let (cid:36) i : Γ i × G i → T for i ∈ [ n ] := { , . . . , n } be Pontryagin dualities with correspondingFourier operators (cid:70) i : L ( G i ) → C (Γ i ) . Writing the product dualityas (cid:36) : Γ × G → T with G := G ⊕ · · · ⊕ G n and Γ := Γ ⊕ · · · ⊕ Γ n , theinduced Fourier operator (cid:70) ⊗· · ·⊗ (cid:70) n on the tensor product is denotedby (cid:70) : L ( G ) → C (Γ) . For any a ⊆ [ n ] with complement a (cid:48) ⊆ [ n ] andany i ∈ [ n ] , we consider the hybrid groups F a := (cid:77) j ∈ [ n ] F j , F a ( i ) := (cid:77) j ∈ [ n ] \{ i } F j with F j = (cid:40) G j if j ∈ a , Γ j if j ∈ a (cid:48) ,which clearly satisfy F a ∼ = F a ( i ) ⊕ F i . This yields C F a ∼ −→ ( C F i ) F a ( i ) as currying isomorphism, which we write f (cid:55)→ f i . Then we definethe hybrid function spaces LC a as the set of all f : F a → C such that f i ( z ) ∈ L ( G i ) for all i ∈ a , z ∈ F a ( i ) and f i ( z ) ∈ C (Γ i ) for all i ∈ a (cid:48) , z ∈ F a ( i ) . Note that LC [ n ] = L ( G ) and LC ∅ = C (Γ) . Given i ∈ a , weset (cid:70) (cid:48) i : LC a → LC a \{ i } by (cid:70) (cid:48) i ( f )( z ) := (cid:70) ( f i ( z )) . Then we have(25) (cid:70) = (cid:70) (cid:48) σ ◦ . . . ◦ (cid:70) σ n for all permutations σ ∈ S n , by Fubini’s theorem as quoted above. Inpractice, one often selects special variables x , . . . , x n ranging over thepositions groups G , . . . , G n and x = ( x , . . . , x n ) ranging over G ; thenone can use (cid:70) x i T := (cid:70) i ( x i (cid:55)→ T ) and (cid:70) x T := (cid:70) ( x (cid:55)→ T ) like quantifierson terms T containing free occurrences of x , . . . , x n . These conventionsare similar to those for differential and integral operators. Example 40.
At this point, it may be useful to review the four mostimportant incarnations of Pontryagin duality—the standard Fourieroperators of analysis (note that (25) is applicable in each of these cases):(a) The classical
Fourier integral (FI) arises when considering the du-ality given by the standard vector duality (cid:104) G | Γ (cid:105) = (cid:104) R n | R n (cid:105) of Ex-ample 9, where (cid:104) ξ | x (cid:105) = e iτx · ξ . In this case, we have(26) (cid:70) s ( ξ ) = (cid:90) R n e iτx · ξ s ( x ) dx ∗ for the Fourier transform. (See the remarks in Example 53a on thetopic of alternative normalizations.)In this context, the mapping property (cid:70) : L ( G ) → C (Γ) isknown as the Riemann-Lebesgue lemma [105, Prop. 6.6.1], in par-ticular the fact that ˆ s ( ξ ) → as | ξ | → ∞ . Moreover, the homo-morphism property (cid:70) ( s (cid:63) s (cid:48) ) = (cid:70) s · (cid:70) s (cid:48) is called the convolutiontheorem . Writing s a ( x ) := s ( x + a ) for the translates of a signal s by an offset a ∈ R n , the two equivariance properties(27) (cid:70) x (cid:0) s ( x + a ) (cid:1) = e iτa · ξ ˆ s ( ξ ) and (cid:70) x (cid:0) e iτx · α s ( x ) (cid:1) = ˆ s ( ξ − α ) are known as the shift theorem and the modulation theorem , respec-tively [24, §6] since translations are obviously also called shifts whilemultiplying with exponentials e iτa · ξ and e iτx · α is known as modulat-ing in engineering parlance. More precisely, this would be frequencymodulation (FM): Taking a sinusoidal signal s ν ( x ) = e iτx · ν of fre-quency ν , one obtains the modulated signal e iτx · α s ν ( x ) = s ν + α ( x ) with altered frequency ν + α .(b) Taking the conjugate torus duality (cid:104) Γ | G (cid:105) = (cid:104) Z n | T n (cid:105) of Example 6for the Pontryagin duality, we obtain Fourier series (FS). Moreprecisely, the multivariate sequence (cid:70) s ( ξ ) ξ ∈ Z n formed by the so-called Fourier coefficients(28) (cid:70) s ( ξ ) = (cid:90) I n e iτx · ξ s ( x ) dx will be seen to constitute the Fourier series of s ; see Example 53b.Here we identify signals s ∈ L ( T n ) with periodic functions de-fined on I n rather than the more usual [0 , τ ] n , where the changeof variables y = τ x yields an additional factor τ − n . The advan-tage of this choice is to achieve a more uniform expression for theFourier transformation: One sees immediately that (26) and (28)differ only in their integration bounds. Correlated to these sig-nals, their spectral space C (Γ) is the space c ( Z n ) of multivariatenull sequences Z n → C ; this is again an instance of the Riemann-Lebesgue lemma [59, Cor. 6.45]. Note, however, that (cid:70) is injectivebut not surjective [59, p. 547]. There are again convolution, shiftand modulation theorems.(c) Interchanging the roles of position and momenta, we obtain the torus duality (cid:104) Γ | G (cid:105) = (cid:104) T n | Z n (cid:105) ; its associated Fourier transformis called [16, §18.5] the discrete-time Fourier transform (DTFT), While s ν (cid:54)∈ L ( R n ) is technically not a signal in our present setting, it can beapproximated by L signals. N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 51 with the corresponding Fourier operator (cid:70) : l ( Z n ) → C ( T n ) givenby(29) (cid:70) s ( ξ ) = (cid:88) x ∈ Z n e iτx · ξ s ( x ) , which may also be viewed as a discretized version of the Fourierintegral (26). In this case, the Riemann-Lebesgue lemma is void (asthe torus T n is compact every continuous function vanishes at infin-ity). Of course one has the usual convolution, shift and modulationtheorems. Note that convolution takes its usual form by writingthe sequences s ∈ l ( Z n ) as multivariate series (cid:80) x ∈ Z n s n x n .At this point it should also be clear why it makes sense—from apurely mathematical point of view—to distinguish the two “mirrorimages” of the torus duality (confer Example 6): We obtain dif-ferent Fourier operators L ( T n ) → c ( Z n ) and l ( Z n ) → C ( T n ) ,whose mapping spaces are obviously quite different. It is only onsuitable subspaces that we may subsequently identify them as es-sentially inverse to each other (Example 53c), linking continuousperiodic with discrete aperiodic signals (see below).(d) Finally, let us take the conjugate cyclic duality (cid:104) Γ | G (cid:105) = (cid:104) Z nN | T nN (cid:105) from Example 11. We recall that both Z N = { , . . . , N − } and T N = N − Z N are the cyclic group Z /N , but while T N (cid:44) → T isnaturally embedded, its dual partner is canonically endowed witha projection Z (cid:16) Z N . In this case, (24) will be the discrete Fourierseries (DFS) given in detail by(30) (cid:70) s ( ξ ) = 1 N n (cid:88) x ∈ T nN e iτx · ξ s ( x ) where ξ ranges over Z nN . It is obvious that (30) is the uniformlysampled form of the Fourier coefficient (28) associated with Fourierseries. Note that the factor /N n arises here from discretizing theconstituent integrals of (28) via (cid:114) . . . dx i (cid:32) (cid:80) x i ∈ T N . . . N − ; thisis in harmony with the chosen normalization of the Haar measure(see the concluding remark in Example 11). It is well known thatuniform sampling in one domain corresponds to periodic repetitionin the other [90, §7.4], so the resulting spectrum under (28) willbe determined by its values on ξ ∈ Z nN . Altogether we obtain atransform of type (cid:70) : L ( T nN ) → C ( Z nN ) . ∗ Changing to the cyclic duality (cid:104) T nN | Z nN (cid:105) , we obtain now the dis-crete Fourier transform (DFT) given by(31) (cid:70) s ( ξ ) = (cid:88) x ∈ Z nN e iτx · ξ s ( x ) , with ξ ranging over T nN . We may view (31) as a sampled form ofthe discrete-time Fourier transform (29). Restricting the latter tosignals s ∈ L ( Z n ) supported within { , . . . , N − } n ⊂ Z n , theinfinite series (29) collapses to the finite sum (31). The result-ing spectrum (cid:70) s ∈ C ( T n ) is subsequently sampled at unit-rootcoordinates T nN ⊂ T n as these are sufficient to reconstruct the sig-nal: The original signal s is again periodically replicated becauseof the uniform sampling of (cid:70) s , but due to the support hypothesisno aliasing occurs and exact reconstruction is ensured. It shouldbe noted, however, that the original signal s has now been iden-tified as periodic , which is inconsistent with our prior assumptionof finite support. The contradiction arises only from the groupstructure—on the set level, we are free to choose between inter-preting complex tuples as representing periodic signals (as for theDFS) or finite signals (as for the DFT).Indeed, all the spaces L ( Z nN ) , L ( T nN ) , C ( Z nN ) , C ( T nN ) are infact the same plain vector space ( C N ) n , and the transformationis the tensor power (cid:70) ⊗ n of a linear map (cid:70) : C N → C N . Up toscaling, the matrix of (cid:70) with respect to the canonical basis is [49,Thm. 39.2] the Vandermonde matrix generated by the N -th rootsof unity T N ⊂ C . This is the form commonly used [16, §16.2] forthe DFS or DFT, with plain integer tuples k, l ∈ { , . . . , N − } n in the exponential e iτ ( k · l ) /N mentioned earlier (7).For physical signals (where x is time and hence ξ is frequency), the char-acteristics of the four transform types may be read off from their do-mains and codomains: Signals on the compact domains T n , Z nN are con-sidered periodic , those on the noncompact domains R n , Z n accordingly aperiodic . In a similar fashion, signals on the discrete domains Z n , Z nN are of course called discrete , those on the nondiscrete domains R n , T n accordingly continuous . Using this terminology, the various Fourieroperators are classified in Figure 3, where the box around each domainsignifies a suitable function space (like L or C ), and the labels onthe arrows refer to the corresponding Fourier transform (using the ab-breviations given in the text above). Similar diagrams are often foundin the pertinent literature; see for example Table 5.3 in [82, p. 396] orFigure 8.2 in [104, p. 145]. N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 53
CONTINUOUS DISCRETEAPERIODIC FI (cid:8)
R Z
DTFT (cid:115) (cid:115)
PERIODIC T FS (cid:51) (cid:51) Z N (cid:9) DFT
Figure 3.
Classical Fourier OperatorsNote that in Figure 3 we have chosen to conflate DFS and DFT ,using the latter term for both. As mentioned earlier (Example 11),their underlying dualities are in fact the same apart from the inessen-tial normalization factor. So from a purely mathematical viewpoint,no distinction is needed. (As alluded to above, one might even argueagainst finite-duration signals as inconsistent with the underlying groupstructure.) Many texts on digital signal processing such as [104] havetherefore also chosen to neglect the difference. As we have seen, thereare nevertheless strong physical arguments in favor of upholding the dis-tinction between discretized bounded but aperiodic signals (construedover Z nN in our setting) and discretized periodic signals (correspond-ingly construed over T nN ); some sources such as [81] or [104] follow thisline. Indeed, the former has even chosen to adapt the normalizations sothat the DFS formula [81, (8.11/12)] becomes identical with the corre-sponding DFT one [81, (8.65/66)], except for the truncation enforcedin the latter. (These formulae also include the inverse transformations,which we will encounter in Example 53d.) // It should also be mentioned that the classical Fourier doublet L (cid:104) Γ | G (cid:105) for a Pontryagin duality (cid:36) : Γ × G → T can be extended via the mea-sure algebra M ( G ) ⊇ L ( G ) consisting of all bounded regular Borelmeasure on G . It is a standard fact [100, Cor. 1.3.2] of Fourier anal-ysis that M ( G ) is indeed a unital algebra (actually a Banach algebra)over C . The Fourier transform can be extended from L ( G ) to M ( G ) with values in the unital algebra BC (Γ) of bounded and uniformly con-tinuous functions, and the resulting map (cid:70) : M ( G ) → BC (Γ) is againa homomorphism that represent the natural action of H ( (cid:36) ) ; see [100,Thm. 1.3.3] and its proof. It is usually called the Fourier-Stieltjes trans-formation . Moreover, it is straightforward [100, §1.3.4] that L ( G ) is asubalgebra of M ( G ) , while BC (Γ) is clearly a superalgebra of C (Γ) .Taken together, this yields the measure doublet [ (cid:70) : M ( G ) (cid:55)→ BC (Γ)] . ∗ Proposition 41.
Let G and Γ be LCA groups under Pontryagin duality (cid:36) : Γ × G → T . Then the measure doublet [ (cid:70) : M ( G ) (cid:55)→ BC (Γ)] is anextension doublet of L (cid:104) Γ | G (cid:105) . Example 42.
The discrete Fourier transform generalizes to the Nichol-son duality ν : ˆ G × G → R ∗ mentioned in Example 12. Under the hy-potheses stipulated there, we have the identification Hom(
G, R ∗ ) ∼ = R n ,which then leads [79, (3.11)] to define (cid:70) : R [ G ] → R n , s (cid:55)→ ˆ s by(32) ˆ s ( ξ ) = (cid:88) x ∈ G (cid:104) ξ | x (cid:105) s ( x ) . It will be noted that this specializes to the discrete Fourier trans-form (31). By the fundamental theorem of finite abelian groups, G decomposes as a product of cyclic groups; thus (32) may be viewedas a multivariate DFT. Additionally, (32) specializes to the Gelfandtransform when taking R = C .The Heisenberg action is defined just as in the Pontryagin case. Infact, the whole setting is almost subsumed by Theorem 37, the onlydifference being the torus R ∗ (cid:54) = T . It is easily checked that everythingnevertheless goes through, so that [ (cid:70) : R [ G ] (cid:55)→ R n ] is indeed a Fourierdoublet (note that Definition 18 allows for Heisenberg algebras overrings). // Remark 43.
Insisting on the larger category of nilquadratic ratherthan Heisenberg groups, one can resort to the symplectic Fourier trans-form [32, Def. 6.6], [38, p. 7]. In this case, one would work with asymplectic group P without a Lagrangian splitting. For example, inthe classical case of P = R n , the symplectic Fourier transform is givenby (cid:70) s ( x, ξ ) = (cid:90) R n (cid:104) x, ξ | y, η (cid:105) ω s ( y, η ) dy dη for “hybrid signals” s ∈ L ( R n ) that depend on position x ∈ R n as well as momentum ξ ∈ R n . The underlying symplectic duality is (cid:104) x, ξ | y, η (cid:105) ω := (cid:104) η | x (cid:105) β / (cid:104) ξ | y (cid:105) β , where (cid:104) ξ | x (cid:105) β = e iτx · ξ is the standardvector duality of Example 9. This is the multiplicative symplecticform corresponding to the (additive) canonical symplectic form Ω G for G = R n under the standard character χ : R → T . See Example 15for its relation to the little Heisenberg groups [ H ] ω and H ( β ω ) .If ι n : R n → R n and π n : R n → R n are, respectively, the standardinjections and projections of the direct sum decomposition of the phase N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 55 space P = R n ⊕ R n , one obtains the following commutative diagram: L ( R n ) (cid:70) (cid:47) (cid:47) ι ∗ n (cid:15) (cid:15) C ( R n ) L ( R n ) (cid:70) (cid:47) (cid:47) C ( R n ) π ∗ n (cid:79) (cid:79) This allows one to recast the standard Fourier transform in termsof the symplectic one. Conversely, the symplectic Fourier transformmay be recovered from the standard Fourier transform (cid:70) on L ( R n ) via (cid:70) = (cid:70) ◦ J ∗ , where J : R n → R n is the canonical symplectic ma-trix (cid:0) I n − I n (cid:1) . Unlike its more common counterpart (cid:70) , the symplecticFourier transform (cid:70) is involutive.All these ideas generalize to the setting of arbitrary LCA groups [51,Ex. 5.2v, p. 26]. Having a nilquadratic extension E : T (cid:26) H (cid:16) P withcommutator form ω := ω E , one sets (cid:70) s ( z ) = (cid:114) P ω ( z, w ) s ( w ) dw .Choosing a Lagrangian splitting P = G ⊕ Γ induces the split exactsequence G ι (cid:26) P π (cid:16) Γ and a factorization (cid:70) = π ∗ (cid:70) ι ∗ generalizing theabove diagram. Since the direct sum P is an LCA group [76, p. 362], ithas its own Fourier operator (cid:70) , and one may check that (cid:70) = (cid:70) ◦ J ∗ with J : P → P defined as in the special case above. (cid:125) Classical Pontryagin duality not only provides the prototypical ex-ample of a Fourier doublet, it also gives rise to an important classof Fourier morphisms in the following way. Recall first that every topological automorphism (= homeomorphism + homomorphism) ofan LCA group G is associated with a unique positive number knownas the modulus δ A of the given automorphism A , and the association Aut( G ) → R > , A (cid:55)→ δ A is a group homomorphism [78, Prop. 17]. Thebest known case is when the group is the vector space R n and theautomorphism is an invertible matrix A ∈ R n × n ; then the modulusis just δ A = | det( A ) | ; see Example 2 of [78, p. 84]. Every automor-phism induces a contravariant action L ( G ) → L ( G ) via the pull-back A ∗ s := s ◦ A and a covariant action C (Γ) → C (Γ) sending σ tothe map A ∗ σ given by ξ (cid:55)→ σ ( A ∗ ξ ) . (We write the pullback of charac-ters in the same way as that of L ( G ) functions since they are definedanalogously.) Proposition 44.
Let G and Γ be LCA groups under Pontryagin duality (cid:36) : G × Γ → T . Then every topological automorphism A : G → G givesrise to a Fourier automorphism ( a, α ) of [ (cid:70) : L ( G ) (cid:55)→ C (Γ)] having ∗ signal and spectral maps a : L ( G ) → L ( G ) , s (cid:55)→ δ A A ∗ s,α : C (Γ) → C (Γ) , σ (cid:55)→ A − ∗ σ, where δ A > is the modulus of the topological automorphism A .Proof. We have to verify (cid:70) ( s ◦ A )( ξ ) = δ − A (cid:70) s ( ξ ◦ A − ) for s ∈ L ( G ) and ξ ∈ Γ . According to (24), the left-hand side is given by (cid:114) G (cid:104)− y | ξ (cid:105) s ( A ( y )) dy = (cid:114) G (cid:104)− A ( y ) | ξ ◦ A − (cid:105) s ( A ( y )) dy = δ − A (cid:114) G (cid:104) y | ξ ◦ A − (cid:105) s ( y ) dy = δ − A (cid:70) s ( ξ ◦ A − ) , where the second equality uses [78, Prop. II.16]. For checking that a isan endomorphism on (cid:0) L ( G ) , (cid:63) (cid:1) , one appeals to [78, Prop. II.16] again,and one sees immediately that α is an endomorphism on (cid:0) C (Γ) , · (cid:1) since A − ∗ effects a substitution. It is clear that both a and α are bijective,hence ( a, α ) is an automorphism of [ (cid:70) : L ( G ) (cid:55)→ C (Γ)] . (cid:3) Coming back to the vector space case G = R n of Proposition 44, thesimplest example of an automorphism is the action of a nonvanishingscalar a ∈ R . Writing S a for both induced actions on signals andspectra, we arrive at the famous similarity theorem (cid:70) S a = | a | − S /a (cid:70) of the classical Fourier transform [24, p. 108]. Remark 45.
The setting of Pontryagin duality (cid:36) : G × Γ → T withits Fourier transform (cid:70) : L ( G ) → C (Γ) also provides a sort of integraloperator (cid:117) : (cid:0) L ( G ) , (cid:63) (cid:1) → C acting as s (cid:55)→ (cid:114) G s ( x ) dx . This is a C -algebra homomorphism since we have (cid:117) = e ◦ (cid:70) , where the evaluation e : (cid:0) C (Γ) , · (cid:1) → C with σ (cid:55)→ σ (0) is itself a homomorphism. (Theassociated initialization i := 1 C (Γ) − e acts as a “deletion operator”when applied to Γ = Z n .)3.3. Fourier Inversion.
In classical as well as abstract harmonic anal-ysis, Fourier operators are always injective : As we shall soon show, theoperator (cid:70) ∧ as well as (cid:70) ∨ in Theorem 37 is in fact a monomorphism.Therefore it is reasonable to try and adapt the function spaces in somesuitable way so as to obtain a bijective Fourier operator . Definition 46. A Fourier doublet (cid:68) = [ (cid:70) : S (cid:55)→ Σ] over a duality β is called regular if the Fourier operator (cid:70) is bijective. Otherwise, thedoublet (cid:68) is called singular .And for avoiding cumbersome terminology, we shall from now on takethe liberty of abbreviating the term “Fourier doublet” by just doublet ,in particular when qualifying it by terms such as regular/singular or N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 57 slain/plain/twain. The same applies to the term “Fourier singlet” tobe introduced later in this section (Definition 52).As with many other algebraic structures, a bijective Fourier operator (cid:70) : S (cid:55)→ Σ is automatically an isomorphism in the appropriate sense:Its inverse ˜ (cid:70) is then a Heisenberg morphism Σ ∧ → S of left Heisenbergmodules over β , or equivalently a Heisenberg morphism Σ → S ∧ of rightHeisenberg modules over β . Moreover, ˜ (cid:70) also respects the respectiveproduct(s) in the case of plain/twain algebras S, Σ . It is then natural todenote this situation by ˜ (cid:70) : Σ (cid:55)→ S . While all this pertains to forwardoperators (cid:70) = (cid:70) ∧ , analogous statements obviously hold for backwardoperators (cid:70) ∨ , using respectively S ∨ , Σ ∨ in place of S ∧ , Σ ∧ .Under Pontryagin duality (cid:36) : Γ × G → T , one immediately obtainsa regular doublet by restricting the codomain of the Fourier operatorto the so-called Fourier algebra A (Γ) := (cid:70) L ( G ) ≤ C (Γ) as in [100,§1.2.3]. Keeping the same notation for the restricted operator, we havea regular doublet [ (cid:70) : L ( G ) (cid:55)→ A (Γ)] under the hypotheses of Theo-rem 37. While this is algebraically trivial, it should be kept in mindthat it is a difficult problem to find a suitable analytic description of theFourier group. While general characterizations remain elusive, thereare important results for certain classes of LCA groups such as [101].There is an analogous construction for the measure algebra M ( G ) ofPropostion 41. Following [100, §1.3.3], we denote the image of M ( G ) under the Fourier-Stieltjes transformation (cid:70) by B (Γ) ≤ BC (Γ) . Thisso-called the Fourier-Stieltjes algebra is unital, and one obtains a reg-ular doublet [ (cid:70) : M ( G ) (cid:55)→ B (Γ)] since (cid:70) is injective [100, Thm. 1.3.6].In applications, this doublet is not very important since the inclusion ofdistributions such as Dirac measures δ a ∈ M ( G ) is preferrably achievedvia tempered distributions (see Example 60 below). For the theoreticaldevelopment, however, this doublet is important because of results suchas Bochner’s characterization of positive-definite functions as Fourier-Stieltjes transforms of nonnegative measures [100, Thm. 1.4.3].The Fourier algebra gives rise to just a regular plain doublet, but itmay be restricted further to obtain a regular twain doublet . These factsappear to be well-known in analysis folklore, though proper referencesare difficult to find. We follow here the hints given in [105, Ex. 6.4.5].Assume (cid:36) : Γ × G → T is a Pontryagin duality with the classical Fourierdoublet [ (cid:70) (cid:36) : L ( G ) (cid:55)→ C (Γ)] over (cid:36) in Theorem 37. Then we mayform the function space(33) L / ( G, Γ) := L ( G ) ∩ (cid:70) − (cid:36) L (Γ) = { s ∈ L ( G ) | ˆ s ∈ L (Γ) } as a subspace of L ( G ) . Here the space L (Γ) in (33) is of coursedefined via the conjugate duality (cid:36) (cid:48) : G × Γ → T . Reversing the roles ∗ of G and Γ and that of (cid:36) and (cid:36) (cid:48) in (33), we obtain L / (Γ , G ) asa subspace of L (Γ) , but now with the Fourier operator (cid:70) (cid:36) (cid:48) of thecorresponding doublet [ (cid:70) (cid:36) (cid:48) : L (Γ) (cid:55)→ C ( G )] over (cid:36) (cid:48) . As we shallsee presently, the original Fourier operator restricts to a twain doublet [ (cid:70) (cid:36) : L / ( G, Γ) (cid:55)→ L / (Γ , G )] such that (cid:70) (cid:36) (cid:48) restricts to a sign inverseof (cid:70) (cid:36) . We call L / (cid:104) Γ | G (cid:105) := [ (cid:70) (cid:36) : L / ( G, Γ) (cid:55)→ L / (Γ , G )] the classical twain doublet since it is cut out from the two classicalFourier doublets [ (cid:70) (cid:36) : L ( G ) (cid:55)→ C (Γ)] and [ (cid:70) (cid:36) (cid:48) : L (Γ) (cid:55)→ C ( G )] . Lemma 47.
Let G and Γ be LCA groups under Pontryagin duality (cid:36) : Γ × G → T . Then we have L / ( G, Γ) = L ( G ) ∩ A ( G ) and theinclusion L / ( G, Γ) ⊆ L ( G ) , which is in general strict.Proof. Note that A ( G ) = (cid:70) (cid:36) (cid:48) L (Γ) is the Fourier algebra, now on thesignal side. We have L / ( G, Γ) ⊆ L ( G ) ∩ (cid:70) (cid:36) (cid:48) L (Γ) since the Fourierinversion theorem [37, (4.32)] ensures s = (cid:70) (cid:36) (cid:48) (cid:80)(cid:70) (cid:36) s ∈ (cid:70) (cid:36) (cid:48) L (Γ) when-ever s ∈ L / ( G, Γ) . For the converse, assume s = (cid:70) (cid:36) (cid:48) σ ∈ L ( G ) for σ ∈ L (Γ) . By virtue of the same inversion theorem (applied on theother side), we have (cid:80) σ = (cid:70) (cid:36) s and hence s ∈ (cid:70) − (cid:36) L (Γ) . This estab-lishes the other inclusion and thus L / ( G, Γ) = L ( G ) ∩ (cid:70) (cid:36) (cid:48) L (Γ) .We prove now L / ( G, Γ) ⊆ L ( G ) . Given s ∈ L / ( G, Γ) , we have s ∈ L ( G ) and s ∈ (cid:70) (cid:36) (cid:48) L (Γ) by the identity just proved. Then weobtain s ∈ C ( G ) ⊂ L ∞ ( G ) by the fundamental mapping property (cid:70) (cid:36) (cid:48) : L (Γ) → C (Γ) of the Fourier transform, also known as abstractRiemann-Lebesgue lemma [100, Thm. 1.2.4a]. From s ∈ L ( G ) ∩ L ∞ ( G ) we have || s || = (cid:114) G | s | dx ≤ || s || ∞ (cid:114) G | s | dx = || s || ∞ || s || < ∞ and thus s ∈ L ( G ) . For the important special case L / ( G, Γ) ⊆ L ( G ) with G = R , see also [105, Prop. 6.4.1].For seeing that the inclusion is in general strict, consider the nor-malized cardinal sine function, sinc x := sin πxπx . We have sinc ∈ L ( R ) ;in fact, (cid:114) R sinc x dx = 1 . Moreover, sinc is also integrable, again withthe value (cid:114) R sinc x dx = 1 . However, it is not absolutely integrable since (cid:114) R | sinc x | dx = ∞ . Hence sinc (cid:54)∈ L / ( R ) ⊆ L ( R ) . (cid:3) This lemma implies that the restrictions (cid:70) (cid:36) : L / ( G, Γ) → L / (Γ , G ) and (cid:70) (cid:36) (cid:48) : L / (Γ , G ) → L / ( G, Γ) are well-defined. We shall now provethat L / (cid:104) Γ | G (cid:105) is a subdoublet of L (cid:104) Γ | G (cid:105) having the alleged properties.Here we define the notion of a subdoublet [ (cid:70) (cid:48) : S (cid:48) (cid:55)→ Σ (cid:48) ] of a doublet [ (cid:70) : S (cid:55)→ Σ] by requiring the inclusions i : S (cid:48) (cid:44) → S and ι : Σ (cid:48) (cid:44) → Σ to N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 59 constitute a Fourier morphism ( i, ι ) . In other words, we have S (cid:48) ⊆ S and Σ (cid:48) ⊆ Σ , and the Fourier operator (cid:70) : S → Σ restricts to the Fourieroperator (cid:70) (cid:48) : S (cid:48) → Σ (cid:48) .The twain algebra structure on L / ( G, Γ) ⊂ L ( G ) ∩ C ( G ) comprisesthe products (cid:63) and · inherited from L ( G ) and C ( G ) , respectively; forthe twain algebra L / (Γ , G ) the situation is analogous but now · is thefirst and (cid:63) the second product. Thus for (cid:70) (cid:36) to be Fourier operator, itmust be a twain homomorphism (cid:70) (cid:36) : (cid:0) L / ( G, Γ) , (cid:63), · (cid:1) → (cid:0) L / (Γ , G ) , · , (cid:63) (cid:1) . that respects the Heisenberg action. However, in claiming the subdou-blet relation L / (cid:104) G | Γ (cid:105) ⊆ L (cid:104) G | Γ (cid:105) we take L / (cid:104) Γ | G (cid:105) as a plain doublet(discarding the second products on both sides). Thus we do not thinkof L (cid:104) G | Γ (cid:105) as a twain doublet with trivial second product. In otherwords, we think of it as a plain subdoublet and not as a twain subdou-blet (as in computer science—prefering downcast to upcast). Let usnow prove these claims. Proposition 48.
Let G and Γ be LCA groups under Pontryagin duality (cid:36) : G × Γ → T . Then L / (cid:104) G | Γ (cid:105) ⊆ L (cid:104) G | Γ (cid:105) is a regular twain doublet.Proof. In the proof of Lemma 47 we have seen that (cid:70) (cid:36) (cid:48) (cid:70) (cid:36) s = (cid:80) s for s ∈ L / ( G, Γ) . By symmetry, (cid:70) (cid:36) (cid:70) (cid:36) (cid:48) s = (cid:80) σ for σ ∈ L / (Γ , G ) ,which means (cid:70) (cid:36) has (cid:70) (cid:36) (cid:48) (cid:80) = (cid:80)(cid:70) (cid:36) (cid:48) as its inverse.Closure under the action of H ( β ) follows immediately from equiv-ariance of (cid:70) (cid:36) and (cid:70) (cid:36) (cid:48) . By showing (cid:0) L / ( G, Γ) , · (cid:1) ⊆ (cid:0) C ( G ) , · (cid:1) and (cid:0) L / ( G, Γ) , (cid:63) (cid:1) ⊆ (cid:0) L ( G ) , (cid:63) (cid:1) as Heisenberg subalgebras, we will at onceensure the subdoublet relation and the closure conditions for the twaindoublet (the two other closure conditions follow by symmetry).We prove that L / ( G, Γ) is closed under pointwise multiplication ,generalizing the hints for [105, Ex. 6.4.5]. Let s, s (cid:48) ∈ L / ( G, Γ) bearbitrary. Then s, s (cid:48) ∈ L ( G ) by Lemma 47, so that Hölder’s inequal-ity [59, Thm. 9.2] yields || ss (cid:48) || ≤ || s || || s (cid:48) || < ∞ and thus ss (cid:48) ∈ L ( G ) .On the other hand, s, s (cid:48) ∈ (cid:70) (cid:36) (cid:48) L (Γ) means we may write s = (cid:70) (cid:36) (cid:48) σ and s (cid:48) = (cid:70) (cid:36) (cid:48) σ (cid:48) for σ, σ (cid:48) ∈ L (Γ) . Now the convolution theorem (thefact that (cid:70) (cid:36) (cid:48) : (cid:0) L (Γ) , (cid:63) (cid:1) → (cid:0) C ( G ) , · (cid:1) is a homomorphism, stated inTheorem 37) implies (cid:70) (cid:36) (cid:48) ( σ (cid:63) σ (cid:48) ) = ( (cid:70) (cid:36) (cid:48) σ )( (cid:70) (cid:36) (cid:48) σ (cid:48) ) = ss (cid:48) ∈ (cid:70) (cid:36) (cid:48) L (Γ) .Altogether, we have established ss (cid:48) ∈ L / ( G, Γ) = L ( G ) ∩ (cid:70) (cid:36) (cid:48) L (Γ) .Finally we show that L / ( G, Γ) is also closed under convolution .Taking s, s (cid:48) ∈ L / ( G, Γ) arbitrary, it is obvious that s (cid:63) s (cid:48) ∈ L ( G ) .But we have also (cid:70) (cid:36) ( s (cid:63) s (cid:48) ) = ( (cid:70) (cid:36) s )( (cid:70) (cid:36) s (cid:48) ) ∈ L (Γ) by applying theprevious argument now to (cid:70) (cid:36) s, (cid:70) (cid:36) s (cid:48) ∈ L / (Γ , G ) ⊆ L (Γ) . Thus wehave s (cid:63) s (cid:48) ∈ L ( G ) ∩ (cid:70) − (cid:36) L (Γ) . (cid:3) ∗ Note that both (cid:70) (cid:36) = (cid:70) ∧ (cid:36) and (cid:70) (cid:36) (cid:48) = (cid:70) ∧ (cid:36) (cid:48) in Propostion 48 areforward Fourier operators, which have backward analogs (cid:70) ∨ (cid:36) and (cid:70) ∨ (cid:36) (cid:48) .As it becomes tedious to track all these distinctions, we shall contentourselves with (cid:70) ∧ := (cid:70) ∧ (cid:36) and (cid:70) ∨ := (cid:70) ∨ (cid:36) (cid:48) , which we call the forward and backward Fourier operators of L / (cid:104) G | Γ (cid:105) .These two operators are inverse to each other, as we have seen in theproof of Propostion 48. We transfer the notation introduced after Re-mark 29 to the corresponding images, ˆ s := (cid:70) ∧ ( s ) for the direct trans-form of a signal s ∈ S and ˇ σ := (cid:71) ∨ ( σ ) for the inverse transform of aspectrum σ ∈ Σ ; no confusion is likely to result from this.Under Pontryagin duality, the position group G is discrete iff themoment group Γ is compact [100, Thm. 1.7.3(a)]. In this case, onecan always isolate a distinguished twain doublet within the L / twaindoublet. For describing it, note first that each position a ∈ G inducesa function d a ∈ L ( G ) with d a ( x ) = δ a,x being the Kronecker delta;we write D ( G ) ≤ L ( G ) for the C -linear span of all the d a . On theother hand, the position a ∈ G induces a character χ a ∈ C (Γ) = C (Γ) with χ a ( ξ ) = (cid:104) ξ | a (cid:105) (cid:36) ; we write C (Γ) ≤ C (Γ) for the C -linear span ofthose characters χ a . Proposition 49.
Let G and Γ be LCA groups under Pontryagin duality (cid:36) : G × Γ → T , with G discrete. Then [ D ( G ) (cid:55)→ C (Γ)] is a regulartwain subdoublet of [ (cid:70) : L / ( G, Γ) (cid:55)→ L / (Γ , G )] with (cid:70) d a = χ a .Proof. We have (cid:70) d a ( ξ ) = (cid:80) x ∈ G (cid:104) ξ | x (cid:105) (cid:36) d a,x = (cid:104) ξ | x (cid:105) (cid:36) , hence (cid:70) d a = χ a .Of course we have d a ∈ L ( G ) as noted above. Since (cid:107) χ a (cid:107) ≤ | Γ | ,we have also χ a ∈ L (Γ) and thus indeed d a ∈ L / ( G, Γ) . ApplyingLemma 47 with G and Γ interchanged, we obtain χ a ∈ L / (Γ , G ) since χ a = (cid:70) d a ∈ (cid:70) L ( G ) . Thus we have established [ D ( G ) (cid:55)→ C (Γ)] asa subobject of the L / doublet on the set-theoretic level. It remainsonly to show closure under convolution, pointwise multiplication andthe Heisenberg action. It suffices to check this on one side, say D ( G ) .It is easy so check that d a (cid:63) d b = d a + b and d a · d b = δ a,b d a . Thus weobtain (cid:0) D ( G ) , (cid:63) (cid:1) ∼ = C [ G ] and (cid:0) D ( G ) , · ) ∼ = C (cid:76) G (cid:77) , which ensures closureunder both product structures. Finally, closure under the Heisenbergaction follows since x • d a = d a + x for x ∈ G and ξ • d a = (cid:104) ξ | a (cid:105) (cid:36) d a for ξ ∈ Γ , while the torus acts trivially via the inclusion T ⊂ C . (cid:3) We call [ D ( G ) (cid:55)→ C (Γ)] the character doublet of the Pontryaginduality (cid:36) : G × Γ → T . Note that in this case the action of the discrete Also known as the direct and inverse
Fourier operators in the literature.
N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 61 group G on both signals s ∈ L ( G ) and spectra σ ∈ C (Γ) = C (Γ) isalready contained in the twain algebra structure since x • s = d x (cid:63) s and x • σ = χ x · σ . Furthermore, it also encodes evaluation and theFourier transform since d a • s = s ( a ) d a and χ a (cid:63) σ = χ a • (cid:70) ∨ σ ( a ) .Since χ ∈ C (Γ) ⊆ C (Γ) is a neutral element in (cid:0) C (Γ) , • (cid:1) , suchalgebras are always unital with C (cid:44) → C (Γ) and with unital Fouriertransform (cid:70) ( δ ) = 1 . We shall see an important instance of a characterdoublet in Example 53c.If G fails to be discrete, the d a are in general distributions while thefunctions χ a do not vanish at infinity. In the crucial example G = R ,we get the Dirac deltas d a = δ a ∈ M ( R ) and the oscillating exponen-tials χ a = e a ∈ BC ( R ) with e a ( ξ ) = e iτaξ . As this example suggests,the character doublet [ D ( G ) (cid:55)→ C (Γ)] is no longer a twain subdoublet of [ (cid:70) : L / ( G, Γ) (cid:55)→ L / (Γ , G )] but just a slain subdoublet of the measuredoublet [ (cid:70) : M ( G ) (cid:55)→ BC (Γ)] . We do not pursue these matters here,but we shall need the following application of the character algebra C (Γ) for describing the Heisenberg closure ¯Σ of a subalgebra Σ ≤ C (Γ) , de-fined as the smallest Heisenberg subalgebra of C (Γ) that extends Σ . Itis obtained by tensoring C [ΣΓ] ≤ C (Γ) , the complex algebra generatedby the translates of Σ , with the character space. Proposition 50.
Let G and Γ be LCA groups under Pontryagin duality (cid:36) : Γ × G → T . The Heisenberg closure ¯Σ of a subalgebra Σ ≤ C (Γ) is isomorphic to C (Γ) ⊗ C C [ΣΓ] .Proof. For the purpose of this proof, let us write the translates of a spec-trum σ ∈ Σ by η ∈ Γ as σ η := η • σ so that every element of C [ΣΓ] canbe written as a polynomial in the σ y . It is then clear that C (Γ) C [ΣΓ] ,the C -linear span of all products χ y σ η · · · σ η k k = y • σ η · · · σ η k k , is con-tained in the pointwise algebra C (Γ) since the latter is closed underthe Heisenberg action. We have ( χ y σ η · · · σ η k k ) ( χ ¯ y ¯ σ ¯ η · · · ¯ σ ¯ η l l ) = χ y +¯ y σ η · · · σ η k k ¯ σ ¯ η · · · ¯ σ ¯ η l l , which shows at once closure under the pointwise product and the iso-morphism with C (Γ) ⊗ C Γ[Σ] . Finally, closure under the Heisenbergaction is clear since x • χ y σ η · · · σ η k k = χ x + y σ η · · · σ η k k ,ξ • χ y σ η · · · σ η k k = (cid:104) ξ | y (cid:105) − χ y σ ξ + η · · · σ ξ + η k k , while the torus T ⊂ C acts trivially.We have thus verified that C (Γ) Γ[Σ] ∼ = C (Γ) ⊗ C Γ[Σ] is a Heisenbergsubalgebra of C (Γ) . It is clearly the smallest such since closure under ∗ the poitwise product and the action of Γ ensures inclusion of C [Γ] ,whereas closure under the action of G brings in the characters χ y . (cid:3) Remark 51.
Note that the algebra Σ may be nonuntial. Then C [Γ] and the tensor algebra C (Γ) ⊗ C C [Γ] will also be nonunital, and thelatter does not contain C (Γ) as a subalgebra. We have already notedthis above for the example G = R where we have the oscillating expo-nentials χ a = e a (cid:54)∈ C ( R ) . (cid:125) Before passing to the crucial examples of the classical Fourier op-erators, it will be useful to introduce one final bit of typology for“Fourier structures”: When signal and spectral space are isomorphic,they may be identified so that the
Fourier operator is an endomorphism in Mod K . Definition 52. A Fourier singlet is a Fourier doublet [ (cid:70) : S (cid:55)→ S ] .Of course, any singlet may still be viewed as a doublet (see Ex-ample 40 above) by choosing not to identify signals and spectra evenwhen this is possible: Even when the spaces coincide, one may chooseto make them formally distinct (say, by introducing Σ := S × { } ).Note also that regular doublets may but need not be viewed as singlets.The intention of distinguishing signals from spectra may be the moti-vation behind writing signals as functions of x but spectra as functionsof ξ .While such a distinction may seem pedantic, it should be called tomind that the situation is very similar for linear maps between vectorspaces f : V → W , where the distinction is crucial when it comes toclassification: While homomorphisms are classified for equivalence bythe rank, endomorphisms are classified for similarity by the elementarydivisors. In the case of Fourier structures, the special role of singletswill become apparent when we consider adjunction of new elements(Remark 88). For the moment, it suffices to point out that the classicaltwain doublet L / (cid:104) Γ | G (cid:105) for a Pontryagin duality (cid:36) : Γ × G → T maybe viewed as a singlet exactly in the self-dual case, i.e. when G ∼ = Γ . Example 53.
Returning to the classical Fourier operators of Exam-ple 40, let us review the inverse transformation (cid:70) ∨ : L / (Γ) → L / ( G ) ,where in each case (cid:70) ∧ : L / ( G ) → L / (Γ) as the restriction of thecorresponding Fourier operator (cid:70) : L ( G ) → C (Γ) according to Pro-postion 40. In this context, the defining formulae (26), (28), (29), (31)for (cid:70) ∧ are referred to as the analysis equations , those for (cid:70) ∨ the syn-thesis equations . We shall now state the latter in each of the four casestreated in Example 40. N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 63 (a) For the
Fourier integral (cid:70) ∧ : L / ( R ) → L / ( R ) , the inverse trans-formation is of the same form, except for the negative sign in theexponential so that(34) (cid:70) ∨ σ ( x ) = (cid:90) R n e − iτx · ξ σ ( ξ ) dξ. At this point, it should also be mentioned that different normal-izations for the Fourier integral—corresponding to different nor-malizations of the underlying Haar measure—are in circulation.Some people use instead of the “ordinary frequency” variable ξ the“angular frequency” ω := τ ξ , so the exponential becomes e ix · ω inthe forward transformation (26) and e − ix · ω in the above backwardtransformation (34), along with a scaling factor τ − n in the latter.Since this destroys the unitary character (when viewing it on thecorresponding Hilbert space—see Proposition 54), the scaling fac-tor is evenly distributed as τ − n/ in both forward and backwardtransformation.(b) For the case of Fourier series , the synthesis of the Fourier coef-ficients (28) is just the summation that builds up their “Fourierseries”, namely(35) (cid:70) ∨ σ ( x ) = (cid:88) ξ ∈ Z n e − iτx · ξ σ ( ξ ) . As with the Fourier integral, some modifications are possible. Inparticular, signals may be taken over periodic domains [0 , T ] ∼ = S with other periods T (cid:54) = 1 . In that case, the exponentials in (28)and (35) become e ± iτx · ξ/T with an additional scaling factor of T − n in front of the Fourier coefficients (28). In that case, it can beconsidered suitable to introduce the “angular velocity” ω := τ /T ,thus simplifying the exponentials to e ± ix · ω .(c) The inverse of the discrete-time Fourier transform (29) is(36) (cid:70) ∨ σ ( x ) = (cid:90) I n e − iτx · ξ σ ( ξ ) dξ, which is identical to the synthesis equations (34) of the Fourierintegral except for the different integration domain (which is ac-tually a parametrization of T n ). As with Fourier series, there arestraightforward modifications for adopting different periods.Moreover, the spectra σ ∈ L / ( T n ) may be viewed as functionson n complex variables z , . . . , z n ∈ T ⊂ C . Assuming conver-gence, the n circles T = S may be expanded to annuli A , . . . , A n to produce an analytic function σ : A × · · · × A n → C . Under thisinterpretation, the analysis equation (29) amounts to summing the ∗ multivariate Laurent series while the synthesis equation (36) ef-fects the computation of the Laurent coefficients, essentially viaCauchy’s integral formula. In other words, the Fourier operators (cid:70) ∧ and (cid:70) ∨ coincide with the (bilateral) (cid:90) -transform and its in-verse, possibly up to minor adoptions (as usual there is no agree-ment on the sign in the exponential).As Z n is discrete, there is a twain subdoublet [ D ( Z n ) (cid:55)→ C ( T n )] of [ L / ( Z n , T n ) (cid:55)→ L / ( T n , Z n )] described in Proposition 49. Inour case, we have d a ( x ) = δ a,x = δ a ,x · · · δ a n ,x n , χ a ( ξ ) = ξ a = ξ a · · · ξ a n n , with the group algebra (cid:0) D ( Z n ) , (cid:63) (cid:1) ∼ = C [ Z n ] consisting of Laurentpolynomials (restrictions of the Laurent series mentioned above)and the path algebra (cid:0) C ( T n ) , · (cid:1) ∼ = C (cid:76) Z n (cid:77) having integer n -tuplesas orthogonal idempotents. In its abstract algebraic form, we havealready seen the twain algebra C (cid:76) Z n (cid:77) (cid:62) C [ Z n ] in Example 20.(d) For the discrete Fourier series (30) the inversion is(37) (cid:70) ∨ σ ( x ) = (cid:88) ξ ∈ Z nN e − iτx · ξ σ ( ξ ); for the discrete Fourier transform the analogous expression is(38) (cid:70) ∨ σ ( x ) = 1 N n (cid:88) ξ ∈ T nN e − iτx · ξ σ ( ξ ) . Note that (37) is a truncated version of (35) while (38) is a dis-cretized form of (36); as in the forward DFS transformation (30),the factor N − n arises from (cid:114) . . . dx i (cid:32) (cid:80) x i ∈ T N . . . N − . Need-less to say there are again various conventions for the sign of theexponential and the distribution of the scaling factor. And onemay of course adopt the viewpoint of identifying DFS and DFT asdiscussed at the end of Example 40.The Nicholson version of the Fourier transform in Example 42 isalso invertible. The fact that (cid:70) : (cid:0) R [ G ] , (cid:63) (cid:1) → (cid:0) R n , · (cid:1) is a isomor-phism was exactly the motivation for the conditions imposed on R ,as mentioned in Example 12. The inversion formula is analogousto (38) above, namely (32) with negated sign and a factor | G | − .Note that cases (a) and (d) are self-dual while the dualities of (b)and (c) are conjuages of each other. Thus one may consider the Fourierintegral and the DFT as Fourier singlets. In the former case, this ap-pears to be universally accepted at least in the one-dimensional case N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 65 (nobody wants to distinguish one-dimensional column vectors from one-dimensional row vectors). But for the DFT, the singlet view appearsto be tied up with the abstract interpretation mentioned at the end ofExample 11. Treating the DFT as a doublet corresponds to the dis-tinction between “discrete Fourier series” and “discrete Fourier trans-forms”, corresponding to the two realizations Z N and T N of the abstractgroup Z /N .(Conversely, some might even argue that Fourier series and the discrete-time Fourier transform are “the same” since inversion and parity are“trivial” operations; this appears to be the viewpoint of [37, p. 99],where only (26), (28), (29) of Example 40 are mentioned, or [100,Ex. 1.2.7], who additionally omits the discrete Fourier transform amongthe “classical groups of Fourier analysis”. However, most other sourcesin signal theory, such as [82], [104], [90], do give a separate treatmentof Fourier series and the discrete-time Fourier transform.) // The L functions form another widespread regular doublet, whichenjoys great popularity due to various beneficial properties ensuingfrom the Hilbert space structure of L functions. The price to be paidfor this convenience is a certain impoverishment of the multiplicativestructure as L functions are in general neither closed under convolutionnor under the pointwise product. As a consequence, we obtain only aslain doublet that we call the square-integrable doublet to be denotedby L (cid:104) Γ | G (cid:105) := [ (cid:70) : L ( G ) (cid:55)→ L (Γ)] . Proposition 54.
Let G and Γ be LCA groups under Pontryagin duality (cid:36) : G × Γ → T . Then L (cid:104) Γ | G (cid:105) is a regular slain doublet containing L / (cid:104) Γ | G (cid:105) as subdoublet, with (cid:70) : L ( G ) → L (Γ) unitary.Proof. We have proved in Lemma 47 that L / ( G, Γ) ⊆ L ( G ) and, du-ally, L / (Γ , G ) ⊆ L (Γ) . Regularity and the required closure properitesof L (cid:104) Γ | G (cid:105) are established in most texts on abstract harmonic analy-sis (they follow essentially from the L theory by completion); see forexample [37, (4.25)]. Of course, the L doublet enjoys a wealth of fur-ther properties (in particular unitarity, see below). It should be notedthat, in the L case, the definition (24) does not generally hold on thenose , but must be understood in the sense of a suitable limit. This ispossible since the classical Fourier transform of Theorem 37 restrictsto the dense subspace L ( G ) ∩ L ( G ) of the Hilbert space L ( G ) , andits image is again dense in L ( G ) ; see [100, §1.6.1], where the unitarynature of (cid:70) : L ( G ) → L (Γ) is also established. (cid:3) At this point we can go back to the classial Fourier operators detailedin Example 40, each of which may be interpreted in their corresponding ∗ L setting according to Proposition 54, and this indeed often used. Asmentioned earlier, the Fourier operator can be suitably normalized soas to become a unitary operator between the Hilbert spaces L ( G ) and L (Γ) . In the case of the DFT, it is then necessary to split thefactor /N n occurring in (31) so that it becomes / √ N n in front of bothforward and backward transformation. Note that each of the classicalFourier operators, defined in (26), (28), (29), (31) and reinterpretedas (cid:70) : L ( G ) → L (Γ) , can be understood as taking the scalar productof a signal s ∈ L ( G ) with the exponential e ξ ( x ) := e iτx · ξ . Thinkingof (cid:70) intuitively as a “matrix” with columns indexed by positions x ∈ G and rows indexed by momenta ξ ∈ Γ , its ( x, ξ ) -entry would be givenby the character (cid:104) x | ξ (cid:105) = e iτx · ξ . While this is can be made preciseimmediately in the DFT case (see the remarks in Example 40d forthe traditional form of the matrix), one may employ the machinery ofrigged Hilbert spaces (also known as Gelfand triples) for treating theother cases along these lines with full rigor. Remark 55.
The L setting is also the hub of the representation the-ory of Heisenberg groups. Under the physical interpretation mentionedearlier (Remark 16), this constitutes the standard approach to repre-senting physical observables . Indeed, the Heisenberg group H pol n overthe standard vector duality (cid:104) R n | R n (cid:105) underpins kinematics, both clas-sical (Hamilton’s equations) and quantum (Heisenberg equation): TheSchrödinger representation ρ h : H n → (cid:72) (cid:0) L ( R n ) (cid:1) yields the latter; it isparametrized by the Planck constant h whose so-called semi-classicallimit h → leads to Hamiltonian mechanics. See also [47], [48] formore on the representation theory of H n . If one dislikes the idea ofconstants tending to zero (though one might interpret this as tak-ing place in a hypothetical sequence of universes with progressivelyless significant quantum effects), the framework of Plain Mechanics(p-mechanics) offers an alternative viewpoint [54]: Before specializingto any quantum or classical (or hyperbolic quantum) version, physicalsystems are described in the “plain” setting of the Heisenberg group H n ,using (cid:0) L ( H n ) , (cid:63) ) as the algebra of observables (including in particularthe Hamiltonian). A so-called universal equation rules the evolutionof the system, which transforms to the Heisenberg equation under therepresentation ρ h and to Hamilton’s equations under ρ . In this frame-work, one may develop corresponding brackets [55], notions of state [56]and a detailed mechanical theory [57]. In a newer presentation [58,IV.1], p-mechanics is treated in a more geometric context where theelliptic / parabolic / hyperbolic cases correspond to different numberrings (complex / dual / double numbers). N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 67
It would be nice to reformulate the last results in term of suit-able tori. For example, the classical case should correspond to thetorus T ε := { bε | b ∈ R } of the dual numbers R ε := R [ ε | ε = 0] with the duality (cid:104) R n | R n (cid:105) ε given by (cid:104) x | ξ (cid:105) ε := 1 + ε ( x · ξ ) . Its L rep-resentation (encoded in the L Fourier doublet) should then yield theclassical Hamilton’s equations just as the standard duality yields theHeisenberg equation. In an even more ambitious enterprise, one mighttry to develop a general theory of “universal equations” for a given du-ality that yields classical and quantum kinematics as special cases. Forthis purpose, some tools developed in [4], [5] may be of help.It should be mentioned that the Heisenberg group also affords someother very famous representations, apart from the all-important Schrödingerrepresentation: The phase-space representation is embodied in the sym-plectic Fourier transform (Remark 43). The theta representation is im-portant in algebraic geometry—it is investigated in Mumford’s monu-mental opus [77], which also mentions the phase-space representationon the way [77, Thm. 1.2]. The
Segal-Bargmann representation onFock space invokes a subspace of the entire functions in n variables [38,§1.6]. (cid:125) Example 56.
We come back to the (normalized) cardinal sine functiongiven in the proof of Lemma 47. It is known [24, p. 106] that its Fouriertransform is the (normalized) rectangle function
Π := χ ( − / , / . Thesquared cardinal sine is also important [24, p. 108]; its Fourier trans-form is the (normalized) triangle function ∆( x ) := max(1 −| x | , . Fur-thermore, in the proof of Lemma 47 we have seen that sinc ∈ L ( R ) ,thus sinc ∈ L ( R ) . Since its transform ∆ is clearly in L ( R ) as well,this shows that in fact sinc ∈ L / ( R ) . In summary, we have theFourier pairs [sinc (cid:55)→ Π] ∈ L (cid:104) R | R (cid:105) \ L / (cid:104) R | R (cid:105) , [sinc (cid:55)→ ∆] ∈ L / (cid:104) R | R (cid:105) , both of which are crucial in signal processing applications. In particu-lar, we have sinc ∈ C ( R ) \ L ( R ) whereas clearly Π ∈ L ( R ) \ C ( R ) .Thus the spaces L ( G ) and C ( G ) are generically seen to have nontriv-ial intersection.For another pair of examples, let us turn to probability. The Gauss-ian distribution with Stigler normalization [107] is g ( x ) := e − πx ; thiscorresponds to a variance of /τ . It is an L function with || g || = 1 and the remarkable property of being a fixed point of the Fouriertransformation. Writing l ( x ) := e −| x | for (twice) the Laplace distribu-tion f ( x | , and c ( ξ ) := τξ ) for the Cauchy distribution f ( ξ | , /τ ) , ∗ we obtain the two Fourier pairs [ g (cid:55)→ g ] , [ l (cid:55)→ c ] ∈ L / (cid:104) R | R (cid:105) , figuring prominently in probability theory.In concluding, let us note that the inclusion L / ( G ) ⊂ L ( G ) ∩ C ( G ) is in general strict, though examples and references seem to be hard tocome by. Taking G = Γ = R again, we have clearly s = x (cid:55)→ − Π( x/ x ) − log(1 − x ) ∈ L ( R ) ∩ C ( R ) . It is harder to see that | ˆ s ( ξ ) | ∼ sin( πξ ) πξ log | ξ | as | ξ | → ∞ and thus ˆ s (cid:54)∈ L ( R ) .This also shows that the Heisenberg twain algebra L ( G ) ∩ C ( G ) doesnot in general map to its counterpart L (Γ) ∩ C (Γ) ; it is thus of littleuse and not considered in Fourier analysis. // Example 57. If (cid:36) : Γ × G → T is again a Pontryagin duality, onemay generalize [37, Prop. 4.27] the square-integrable doublet by defin-ing a Fourier transform from L p ( G ) to L q (Γ) for any p ∈ [1 , withHölder conjugate q . Note that q ≥ in this case, so this cannot be asinglet except for p = q = 2 and G ∼ = Γ yielding of course L (cid:104) G | G (cid:105) .Nevertheless, one does get a Fourier doublet [ (cid:70) : L p ( G ) (cid:55)→ L q (Γ)] ingeneral.This is a slain doublet just as in the famous p = q = 2 case. In fact,closure under the pointwise product is immediately seen to break downfrom examples such as s = x (cid:55)→ x − / ∈ L / [0 , with s (cid:54)∈ L / [0 , .As for the convolution product (fixing any particular < p < ), it isknown that closure in L p ( G ) is equivalent to G being discrete, in facteven for nonabelian locally compact groups [2, Prop. 2.1]. Furthermore, (cid:70) : L p ( G ) → L q (Γ) is apparently not surjective , so this is in generalonly a slain singular doublet. // Let us summarize the typology of Fourier structures as follows: Justas for morphisms in any other category, we have isolated the two cru-cial properties of Fourier operators—qua linear maps—as being iso-morphisms (regular versus singular doublets) and endomorphism (sin-glets versus doublets). Furthermore, we distinguish the structures ofslain/plain/twain doublets. We illustrate the various possibilities bylisting for each case some natural example under Pontryagin duality. We point to MathStackExchange https://math.stackexchange.com/questions/67910/a-fourier-transform-of-a-continuous-l1-function for a detailed estimate. See MathStackExchange https://mathoverflow.net/questions/238692/fourier-transform-surjective-on-lp-mathbbrn-for-p-in-1-2 foran argument.
N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 69 (We list only doublets; as mentioned above, one may view them as sin-glets for self-dual groups if desired. But this singlet/doublet distinctionis only important for special purposes such as adjunction of elements.)
Slain Plain TwainSingular [ L / ( G ) (cid:55)→ L (Γ)] [ L ( G ) (cid:55)→ C (Γ)] [ L / ( G ) (cid:55)→ L (Γ) ∩ C (Γ)] Regular [ L ( G ) (cid:55)→ L (Γ)] [ L ( G ) (cid:55)→ A (Γ)] [ L / ( G ) (cid:55)→ L / (Γ)] Table 1.
Examples of Fourier StructuresBefore turning to our most important example from an alogorithmicperspective in the next Section, let us mention a few other Fourierdoublets.3.4.
The Schwartz Class for Pontryagin Duality.
Classical Fourieranalysis comes into its own when introducing the space of
Schwartzfunctions (cid:83) ( R ) and its dual (cid:83) (cid:48) ( R ) , the space of tempered distribu-tions [108]. It is a remarkable fact that even this seeminlgy specialconstruction on real functions can be generalized to the abstract settingof Pontryagin duality, where Schwartz functions are named Schwartz-Bruhat functions, after Laurent Schwartz for the classical theory [103]and François Bruhat for the abstract construction [25]. Following thelatter, one starts from LCA groups G and Γ under Pontryagin dual-ity (cid:36) : G × Γ → T and defines the Schwartz-Bruhat space (cid:83) ( G ) byappealing to the structure theory of LCA groups. We shall only out-line the essential steps of the construction [25, §9], [119, §11]:(1) One starts from the important fact [91, p. 162] that any LCAgroup G is an inverse limit of abelian Lie groups ( G α | α ∈ I ) ,obtained by dualizing [76, Thm. 2.5/Cor. 1] the representa-tion of Γ as a direct limit of its compactly generated subgroups (Γ α | α ∈ I ) . Defining (cid:83) ( G ) as direct limit of (cid:0) (cid:83) ( G α ) | α ∈ I (cid:1) ,it remains to define (cid:83) ( L ) when L is an abelian Lie group (in adifferent terminology: “having no small subgroups”).(2) In that case [76, Thm. 2.4], L is isomorphic to R m ⊕ T n ⊕ D for a discrete group D , which may again be written as a directlimit of its finitely generated subgroups D β so that L is thedirect limit of the elementary groups ( R m ⊕ T n ⊕ D β | β ∈ J ) .Applying primary decomposition on each D β , one obtains therepresentation E β = R m ⊕ T n ⊕ Z k ( β ) ⊕ H β for the elementary A concise summary may also be found on the nlab page https://ncatlab.org/nlab/show/Schwartz-Bruhat+function . ∗ groups, with H β finite. One defines (cid:83) ( L ) as functions that areSchwartz-Bruhat on some E β and vanishing elsewhere.(3) A function f on an elementary group R m ⊕ T n ⊕ Z k ⊕ H is calledSchwartz-Bruhat if f ( − , ξ, − , h ) : R m ⊕ Z k → C is a Schwartzfunction for each ( ξ, h ) ∈ T n ⊕ H . As usual, Schwartz functions R m ⊕ Z k → C are defined as smooth in the variable x ∈ R m ,and with all derivatives ∂ α f /∂x a ( a ∈ N m ) of rapid decay. Afunction ϕ : R m ⊕ Z k → C is said to be of rapid decay if itsatisfies ϕ ( x, ν ) = o ( x − a ν − b ) for all ( a, b ) ∈ N m × N k .The direct limit it Item (1) above can be realized by defining G ϕ → C to be Schwartz-Bruhat iff it factors, for some α , as G π (cid:16) G α ˜ ϕ → C with π the canonical projection onto G α = G/ Ann(Γ α ) and ˜ ϕ anySchwartz-Bruhat function in the sense of Item (2). See [75, Thm. 27]for exchanging subgroups and quotients under dualization.There are two variations of this definition. The first is to retainitem (3) above, but top it by a more intrinsic characterization of differ-entiation in terms of one-parameter subgroups [117]. Using some morefunctional analysis, these ideas can even be generalized to non-abelianlocally compact groups [6]. The second variation is to bypass Lie groupsaltogether, defining (cid:83) ( G ) directly in terms of growth rates [83].We obtain one and the same Schwartz-Bruhat space (cid:83) ( G ) ⊂ L ( G ) with any of these definitions. It turns out that (cid:83) ( G ) is dense in L ( G ) according to [117, Thm. 3.3], and dense in L ( G ) according to [83,p. 42]. Of course, this generalizes well-known facts of classical analysis;they show that (cid:83) ( G ) is in fact “big enough to be useful”. Let us nowrelate the Schwartz-Bruhat space to the classical Fourier singlet andthen review the classical Fourier operators of Example 40. Theorem 58.
Let G and Γ be LCA groups under Pontryagin duality (cid:36) : Γ × G → T . Then [ (cid:70) : (cid:83) ( G ) (cid:55)→ (cid:83) (Γ)] is a regular twain doubletcontained in the classical twain doublet L / (cid:104) Γ | G (cid:105) .Proof. In the above-mentioned treatments of Schwartz-Bruhat spaceit is proved that [117, Thm. 3.2] the (forward) Fourier operator (cid:70) ∧ of L / (cid:104) Γ | G (cid:105) restricts to a map (cid:83) ( G ) → (cid:83) (Γ) . By symmetry, it isalso true that the backward Fourier operator (cid:70) ∨ of L / (cid:104) Γ | G (cid:105) restrictsto (cid:83) (Γ) → (cid:83) ( G ) . Since (cid:70) ∧ and (cid:70) ∨ are mutually inverse in the classicalFourier singlet L / (cid:104) Γ | G (cid:105) , the same must be true for the correspondingrestrictions in (cid:83) ( G ) (cid:55)→ (cid:83) (Γ) .Closure of (cid:83) ( G ) under convolution is established in [83, p. 42], where (cid:83) ( G ) is (temporarily) denoted by (cid:67) ( G ) and the even stronger closure N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 71 property (cid:65) ( G ) (cid:63) (cid:67) ( G ) ⊆ (cid:67) ( G ) for a certain space (cid:65) ( G ) ⊆ L ∞ ( G ) containing (cid:67) ( G ) is stated. Since we have s · s (cid:48) = (cid:70) ∨ ( (cid:70) ∧ s (cid:63) (cid:70) ∧ s (cid:48) ) for s, s (cid:48) ∈ (cid:83) ( G ) ⊆ L / ( G ) , closure under convolution (cid:63) implies closureunder the pointwise product · as well.It is also mentioned in [83, p. 42] that (cid:83) ( G ) = (cid:67) ( G ) is translation-invariant, meaning closed under the action of G ⊂ H ( (cid:36) ) . By symme-try, the same holds for the action of Γ ⊂ H ( (cid:36) ) . Since the torus actionof T ⊂ C is trivial, this shows that (cid:83) ( G ) is closed under the Heisenbergaction. (cid:3) We call (cid:83) (cid:104) Γ | G (cid:105) := [ (cid:70) : (cid:83) ( G ) (cid:55)→ (cid:83) (Γ)] the Schwartz singlet over thegiven Pontryagin duality (cid:36) : G × Γ → T . As stated earlier, this gener-alizes some classical facts that we can now formulate on the basis of theforward and backward Fourier operators (cid:70) ∧ and (cid:70) ∨ of Example 40. Example 59.
Since (cid:70) ∧ : (cid:83) ( G ) → (cid:83) (Γ) and (cid:70) ∨ : (cid:83) (Γ) → (cid:83) ( G ) are re-strictions of the corresponding Fourier operators in the classical Fouriersinglet L (cid:104) Γ | G (cid:105) , we need only review the signal space (cid:83) ( G ) and thespectral space (cid:83) (Γ) in each of the four cases:(a) For the Fourier integral (Example 40a) on G = Γ = R n , we obtainof course the classical Schwartz class (cid:83) ( R n ) mentioned in item (3)above. So we have ϕ ∈ (cid:83) ( R n ) iff ϕ is smooth and of rapid decay,in the sense that ϕ ( x ) = o ( x − α ) for all α ∈ N n .(b) For Fourier series (Example 40b) we have G = T n and Γ = Z n ; inthis case the elements of (cid:83) ( T n ) are just smooth functions (since T n is compact) while those of (cid:83) ( Z n ) are the sequences η : Z n → C ofrapid decay, so η ( k ) = o ( k − α ) for all α ∈ N n .(c) For the discrete-time Fourier transform (Example 40c), the situa-tion is of course the same as for Fourier series, but with the rolesof G and Γ interchanged.(d) For the discrete Fourier transform (Example 40d), we have the fi-nite groups G = T nN and Γ = Z nN , so signal and spectral spaces bothstay the same as in the classical DFT Fourier singlet L / (cid:104) T nN | Z nN (cid:105) because finite sequences are trivially “smooth” and “rapidly decay-ing”, hence we have (cid:83) ( G ) ∼ = (cid:83) (Γ) ∼ = ( C N ) n . The same is of coursetrue for discrete Fourier series.As explained in item (2) above, the elementary groups are composedexactly of the component groups listed above. But the correspondingSchwartz-Bruhat functions must have the prescribed decay of item (3),for all relevant variables jointly . For example, consider the smooth (butnon-analytic) function f : C → R , z = x + iy (cid:55)→ | exp( z ) | = e − x +6 x y − y . ∗ Then f ( x , y ) = O ( e − y ) and f ( x, y ) = O ( e − x ) are in (cid:83) ( R ) for anyfixed x , y ∈ R ; but we have f ( t, t ) = e t → ∞ so that f (cid:54)∈ (cid:83) ( R ) .Similarly, the restriction ˜ f : R × Z → R is in (cid:83) ( R ) when fixing thediscrete variable and in (cid:83) ( Z ) when fixing the continuous variable; nev-ertheless we have again ˜ f (cid:54)∈ (cid:83) ( R × Z ) . This shows that the decayproperties of Schwartz-Bruhat functions on elementary groups, as peritem (3) above, are in general stronger than the corresponding decayproperties for each component group separately. // Having introduced the Schwartz space (cid:83) ( G ) for a locally compactgroup, one may of course define the space of tempered distributions (cid:83) (cid:48) ( G ) in the usual way [117, Def. 4.1], and the induced Fourier transform (cid:70) isagain an automorphism of C -vector spaces [25, p. 61]. Of course, onecannot expect closure under pointwise multiplication (as δ ∈ (cid:83) (cid:48) ( R ) shows) or convolution (as ∈ (cid:83) (cid:48) ( R ) shows). In summary, one obtainsthe following result. Proposition 60.
Let G and Γ be LCA groups under Pontryagin duality (cid:36) : Γ × G → T . Then (cid:70) : (cid:83) (cid:48) ( G ) (cid:55)→ (cid:83) (cid:48) (Γ) yields a regular slain doublet (cid:83) (cid:48) (cid:104) Γ | G (cid:105) containing the Schwartz doublet (cid:83) (cid:104) Γ | G (cid:105) . (cid:3) In fact, one may show [117, Thm. 4.3] that (cid:83) (cid:48) (cid:104) Γ | G (cid:105) contains theregular doublet [ M ( G ) (cid:55)→ BC (Γ)] of Proposition 41.The above statement that (cid:83) (cid:104) Γ | G (cid:105) ≤ (cid:83) (cid:48) (cid:104) Γ | G (cid:105) misses what is oftenperceived as the most crucial property of the pointwise multiplicationand convolution of tempered distributions: While one may not applythese to two arbitrary signals s, s (cid:48) ∈ (cid:83) (cid:48) ( G ) or spectra σ, σ (cid:48) ∈ (cid:83) (cid:48) ( G ) ,it is always possible to apply them to s ∈ (cid:83) ( G ) and s (cid:48) ∈ (cid:83) (cid:48) ( G ) , orto σ ∈ (cid:83) (Γ) and σ (cid:48) ∈ (cid:83) (cid:48) (Γ) .We are thus led to introduce the structure of a twain module ( M, (cid:63), · ) over a twain algebra ( A, (cid:63), · ) , defined as an abelian group ( M, +) to-gether with two scalar multiplications (cid:63) : A × M → M and · : A × M → M such that ( M, (cid:63) ) is a module over ( A, (cid:63) ) while ( M, · ) is a moduleover ( A, · ) . Clearly, we recover modules in the usual sense (“plain mod-ules”) if A is a plain algebra (one of its multiplications being trivial)and naked abelian groups (“slain modules”) if R is a slain algebra (bothmultiplications being trivial).If A is moreover a Heisenberg twain algebra over β , we obtain a Heisenberg twain module M provided that ( M, (cid:63) ) is a recto Heisenbergmodule over β and ( M, · ) is a verso Heisenberg modules over β . Herethe recto/verso distinction is analogous to the case of Heisenberg twainalgebras. Writing the action of both H ( β ) on A and M by juxta-position, ( M, (cid:63) ) being recto thus means x • ( a (cid:63) s ) = ( x • a ) (cid:63) s = a (cid:63) ( x • s ) N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 73 and ξ • ( a (cid:63) s ) = ( ξ • a ) (cid:63) ( ξ • s ) for all a ∈ A, s ∈ M and x ∈ G, ξ ∈ Γ ,where H ( β ) = T G (cid:111) Γ ; for ( M, · ) being verso one just reverses the rolesof scalars and operators.Now let M (cid:48) be another Heisenberg twain module for the same Heisen-berg group H ( β ) such that [ (cid:70) : M (cid:55)→ M (cid:48) ] is a slain doublet and let A (cid:48) be another Heisenberg twain module such that [ (cid:71) : A (cid:55)→ A (cid:48) ] is a twaindoublet. Then we call [ (cid:70) : M (cid:55)→ M (cid:48) ] a twain doublet over [ (cid:71) : A (cid:55)→ A (cid:48) ] if (cid:70) ( a (cid:63) s ) = (cid:71) a · (cid:70) s and (cid:70) ( a · s ) = (cid:71) a (cid:63) (cid:70) s for a ∈ A and s ∈ M . Byabuse of notation, the Fourier operator (cid:71) on A is commonly denotedby (cid:70) as well. We can now capture the observation made above in thefollowing statement. Proposition 61.
Let G and Γ be LCA groups under Pontryagin duality (cid:36) : Γ × G → T . Then (cid:83) (cid:48) (cid:104) Γ | G (cid:105) is a regular Heisenberg twain doubletover the Schwartz doublet (cid:83) (cid:104) Γ | G (cid:105) . (cid:3) Proof.
We have seen that (cid:83) (cid:104) Γ | G (cid:105) is a twain doublet (Theorem 58) andthat (cid:83) (cid:48) (cid:104) Γ | G (cid:105) is a regular slain doublet (Proposition 60). The com-patibility relation required for their Fourier operators follows from theusual duality definitions (using parentheses for the natural pairing),namely ( (cid:70) s | ϕ ) = ( s | (cid:70) ϕ ) , ( a · s | ϕ ) = ( s | a · ϕ ) , ( a (cid:63) s | ϕ ) = ( s | (cid:80) a (cid:63) ϕ ) for a, ϕ ∈ (cid:83) ( G ) and s ∈ (cid:83) (cid:48) ( G ) , using also (cid:70) = (cid:80) . Similar remarkspertain to showing compatibility of the Heisenberg actions, defined by ( x • s | ϕ ) = ( s | ( − x ) • ϕ ) and ( ξ • s | ϕ ) = ( s | ξ • ϕ ) for x ∈ G and ξ ∈ Γ . (cid:3) For applications in analysis it is very important that we can usecertain relations between Fourier and differential operators. Roughlyspeaking, differentiating a signal corresponds to multiplying the spec-trum by a polynomial (the symbol of the differential operator appliedto the signal), and vice versa. It is thus crucial for us to incorpo-rate these relations in our algebraic framework. In doing so, we willnot only capture the classical situation of Schwartz functions (cid:83) ( R n ) and tempered distributions (cid:83) (cid:48) ( R n ) but also their generalizations in the Schwartz-Bruhat setting . Since we are mainly interested in the classicalcase, though, we shall only sketch the general procedure.For a convenient and uniform treatment of differential operatorson (cid:83) ( G ) and thus, by duality, on (cid:83) (cid:48) ( G ) , we refer to the paper [6] al-ready mentioned in §3.4. For handling differential operators in generalLCA groups, one must admit functions that may depend on arbitrar-ily many variables ( x q | q ∈ Q ) . Their cardinality | Q | = dim( G ) canbe uncountable—clearly this is a theoretical seting not a constructiveframework. At any rate, differential operators are described by multi-indices , namely functions µ : Q → N with supp µ = { q ∈ Q | µ q (cid:54) = 0 } ∗ being finite. We follow [6] in denoting the set of all such multi-indicesby N Q . Then a differential operator has the general form(39) T = (cid:88) µ ∈ N Q p µ ∂ µ , where the p µ ∈ C ∞ ( G ) are in the first instance arbitrary smooth func-tions in the sense of Bruhat [6, §1.2], [25, Def. 2]; we shall later restrictthem to polynomials. The differential operator (39) is locally of finiteorder , meaning in a sufficiently small neighborhood of any point in G ,there are only finitely many nonzero summands that contribute.For fleshing this out in some more detail, let us briefly go throughthe development of [6]. For any LCA group G , one can define the Liealgebra
Lie( G ) as its system of one-parameter subgroups [6, §1]. Oneobtains a topological Lie algebra [6, Thm. 1.1] whose dimension co-incides with that of G in case it is finite. Defining a notion of basissuitable for this setting [6, §0], the vector space R Q for arbitrary in-dex sets Q has the usual Kronecker basis, and every other basis hasthe same cardinality | Q | . For the Lie algebra one gets [6, (1.1)] asexpected Lie( G ) ∼ = R Q , thus allowing to fix a basis ( e q | q ∈ Q ) .According to the description outlined in §3.4, the LCA group G maybe viewed as an inverse limit of Lie groups ( G α | α ∈ I ) . It is possibleto realize the latter as G α = G/H α , where λ ( G ) := ( H α | α ∈ I ) formsa decreasing filtration of so-called good subgroups of G . This complieswith the terminology of [25, §1], where compact subgroup are called good if their quotients are Lie groups. Just as with the LCA group G itself, also its Lie algebra Lie( G ) is then an inverse limit of the corre-sponding genuine Lie groups Lie(
G/H α ) ; see [6, (1.5)], keeping in mindthat our present setting is somewhat simpler since G is commutativeand thus an LP group [117, Prop. 1.10ii].For defining the algebra of smooth functions (cid:69) ( G ) ≡ C ∞ ( G ) , it suf-fices [6, §1.2] to define the subalgebra (cid:68) ( G ) of compactly supported ones since each function in (cid:69) ( G ) agrees with a function of (cid:69) ( G ) in asmall neighborhood of any fixed point x ∈ G . One defines first thespace (cid:68) ( G : H α ) of compactly supported smooth functions invarianton a good subgroup H α ≤ G . Functions χ invariant on H α are in bi-jective correspondence with functions ˜ χ on the Lie group G α = G/H α ,so it is natural to declare χ ∈ (cid:68) ( G : H α ) iff ˜ χ ∈ C ∞ ( G α ) . Then therepresentation of G as an inverse limit over λ ( G ) translates [6, (1.6)]into the direct limit(40) (cid:68) ( G ) = (cid:91) α ∈ I (cid:68) ( G : H α ) , N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 75 so ψ ∈ (cid:68) ( G ) iff xH α (cid:55)→ ψ ( x ) is a smooth Lie group function G α → C for some α ∈ I .We explain now the action of the differential operator T in (39) ona smooth function, which equals locally some ψ ∈ (cid:68) ( G ) around apoint x ∈ G . Each one-parameter subgroup u ∈ Lie( G ) induces aderivation ψ (cid:55)→ ψ (cid:48) by the pointwise limit ψ (cid:48) := lim t → (cid:0) u ( t ) • ϕ − ϕ (cid:1) /t ,where • denotes the Heisenberg action of G ≤ H ( (cid:36) ) on (cid:83) ( G ) ≤ L ( G ) .The derivations induced by the basis vectors e i are regarded as partialderivatives and denoted by ∂ i .By iterating and averaging over all possible differentiation orders [6,(2.2)], this is then generalized to higher-order partial derivatives ∂ α for α ∈ N I . Since ψ ∈ (cid:68) ( G ) is invariant on some H α as per (40),with G α = G/H α being a Lie algebra of finite dimension n , one canfind a neighborhood U of x that is invariant under H α and a projectionmap ρ : U → ¯ U ⊆ R n such that ψ ∈ (cid:68) ( G : H α ) iff ψ = ¯ ψ ◦ ρ fora smooth function ¯ ψ : ¯ U → C in the usual sense [6, Lem. 2.13]. Inthe neighborhood U , the action of T is given in terms of the standardpartial derivatives in R n by T ψ = (cid:88) | µ |≤ N p µ ∂ µ ¯ ψ∂q µ ◦ ρ, where the local order N may depend on U .Let us now define the LCA version of the Weyl algebra A G ( C ) asthe operator ring consisting of all those T whose representation (39)involves only polynomial coefficient functions p µ . Here a smooth func-tion G → C is called a polynomial [3, §1] if its restriction to eachcompactly generated closed subgroup C of G can be written as a poly-nomial in a finite collection of real characters on C , here defined justas the characters but with the additive reals instead of T = S as theirtarget group. Remark 62.
The theory of real characters is important for LCA groupsas well as more general topological groups [33] [8]. They are also crucialfor building up the Laplace transformation in the LCA setting [69] [66];see Remark 65 below. Real characters come in three different guises: • As defined above, they may be taken as continuous homomor-phisms G → ( R , +) . Also [44, Def. (24.33)] calls such objects“real characters” while [66, Def. 2] refers to them as “linear func-tionals”. • Sometimes, real characters are characterized as continuous ho-momorphisms G → ( R + , · ) ; this is the stance taken in [69]and [66, Def. 1]. Their objective is to define Laplace transforms ∗ in the LCA setting, where complex characters G → ( C × , · ) takethe role of the usual characters G → T = S ≤ C × . Follow-ing [66] in calling the latter “unitary characters”, it is clear that acomplex character has a unique polar decomposition into a uni-tary character and a real character in the Mackey-Liepins senseof a continous homomorphism G → ( R × , · ) . It is clear thatMackey-Liepins characters and Hewitt-Ross characters may beidentified via log : ( R + , · ) → ( R , +) . • Each real character of G corresponds bijectively [66, Lem. 1] to aone-parameter subgroup of ˆ G ; by definition, this is a continuoushomomorphism ( R , +) → ˆ G .The bijection between real characters and one-parameter subgroupsgoes as follows [44, (24.43)]. Given a real character χ : G → ( R , +) ,the induced one-parameter subgroup ˆ χ : ( R , +) → ˆ G is defined by (cid:104) ˆ χ ( t ) | x (cid:105) = (cid:104) χ ( x ) | t (cid:105) , meaning ˆ χ ( t )( x ) = e iτχ ( x ) t for t ∈ R and x ∈ G .In the notation of [83, §3], this means ˆ χ ( t ) = Exp( tχ ) .Conversely, if ˆ χ : ( R , +) → ˆ G is a one-parameter subgroup, the map t (cid:55)→ (cid:104) ˆ χ ( t ) | x (cid:105) for fixed x ∈ G yields a continuous homomorphism (cid:104) ˆ χ | x (cid:105) : ( R , +) → T and thus a character on ( R , +) . Since the LCAgroup ( R , +) is self-dual [100, Ex. 1.2.7a], the character (cid:104) ˆ χ | x (cid:105) corre-sponds to a unique real number that we take to define χ ( x ) . By thedefinition of the correspondence ˆ R ∼ = R , this yields (cid:104) ˆ χ ( t ) | x (cid:105) = (cid:104) χ ( x ) | t (cid:105) ,which establishes the claimed bijection. (cid:125) In this terminology, Bruhat’s definition of the
Schwartz space meansthat ψ ∈ (cid:83) ( G ) iff T ψ is bounded for each T ∈ A G ( C ) . It is theneasy to see that the Weyl algebra A G ( C ) acts on (cid:83) ( G ) and thus—byduality—also on (cid:83) (cid:48) ( G ) .3.5. Classical Schwartz Functions and the Weyl Algebra.
Weshall from now on focus on the case most important for applications,the standard vector duality (cid:104) R n | R n (cid:105) from Example 9 whose Pontryaginduality (cid:36) is the exponentiated inner product on R n . Thus setting G = Γ = R n , the classical Weyl algebra A G ( C ) = A n ( C ) acts onthe Schwartz class (cid:83) ( R n ) of rapidly decaying functions as well as itsassociated space (cid:83) (cid:48) ( R n ) of tempered distributions.Let us first study the action of A n ( C ) on Schwartz functions, whichis induced by the original Heisenberg action. Thus we are in fact con-fronted with two actions(41) A n ( C ) × (cid:83) ( R n ) → (cid:83) ( R n ) and H ( (cid:36) ) × (cid:83) ( R n ) → (cid:83) ( R n ) . N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 77
Let e ¯ α denote the modulation action of α ∈ Γ ≤ H ( (cid:36) ) , which corre-sponds to multiplication by e ¯ α ( x ) := e iτx ¯ α . If ¯ α is α times the j -thstandard basis vector of R n for some scalar α ∈ R , we write e αj in placeof e ¯ α so that e ¯ α = e α · · · e α n n for general modulations. We write also t ¯ a for the translation action of ¯ a ∈ G ≤ H ( (cid:36) ) so that t ¯ a s ( x ) = s ( x − a ) .In analogy to the modulations, t ak denotes t ¯ a with ¯ a equal to a timesthe k -th basis vector and a ∈ R , hence t ¯ a = t a · · · t a n n for general trans-lations. As a complex algebra, the operators of H ( (cid:36) ) = T G (cid:111) Γ aregenerated by the e αj and t ak . While modulations and translations com-mute amongst each other, they are linked by the crucial Heisenbergrelations e αj t ak = δ jk αa t ak e αj harking back to (H ).It is clear that the noncentral generators e j := e j and t k := t k of theHeisenberg group H ( (cid:36) ) induce, respectively, the generators x j and ∂ k of the Weyl algebra A n ( C ) . In the operator algebra End (cid:83) ( R n ) , each x j commutes with each of the e αj and each ∂ k with each of the t ak . Toavoid awkward iτ factors (in most places), it is customary to intro-duce the scaled partials iτ D k := ∂ k . Then the Heisenberg relationscorrespond to the Weyl relations [ x j , D k ] = δ jk iτ , and we have the cross-relations [ D k , e αj ] = δ kj ξe αj and [ t ak , x j ] = δ jk a j .In more detail, the said correspondence between the generators ofthe H ( (cid:36) ) = TR n (cid:111) R n and A n ( C ) = C (cid:104) ∂, x (cid:105) actions arises fromviewing the latter as the universal enveloping algebra of the Lie al-gebra of the former. Since we have seen in §2.3 that the Heisenbergtwists ˆ J , ˇ J : H ( (cid:36) ) → H ( (cid:36) ) o plays an important role for H ( (cid:36) ) , it isplausible to expect something similar for the induced map on A n ( C ) ,which we shall again denote by ˆ J .On the level of Lie algebras, the induced map is the differential at theunit element, and it is easy to see that as such, ˆ J flips the sign of mo-mentum vectors and fixes position vectors while ˇ J works the other wayround. Hence the corresponding anti-automorphisms A n ( C ) → A n ( C ) are given by ˆ J ( x α D β ) = ( − | β | D β x α and ˇ J ( x α D β ) = ( − | α | D β x α , re-spectively. It is easy to see that these maps are involutions (= involutiveanti-automorphisms, as in §2.3), where ˆ J is known as the (standard)transposition [31, §16.2], [102, §V1a].One can turn both maps into honest automorphisms. Recall from §2.3the identification H ( (cid:36) ) ∼ = H ( (cid:36) ) o given by c ( x, ξ ) ↔ c ( ξ, x ) . The in-duced map on the enveloping algebra yields the corresponding identifi-cation A n ( C ) ∼ = A n ( C ) o with x ↔ ∂ . If we combine it with ˆ J : A n ( C ) → A n ( C ) o and the scaling factor iτ = ∂/D , we obtain the automorphism(42) ˆ f : A n ( C ) → A n ( C ) with x (cid:55)→ D x , D x (cid:55)→ − x ∗ called the Fourier transform on the Weyl algebra [31, §5.2], [102, §V2a].To be precise, we should perhaps call ˆ f the forward Fourier transform,whereas the automorphism ˇ f induced by ˇ J : A n ( C ) → A n ( C ) o wouldbe called the backward Fourier transform—it turns out to be ˆ f − . Onbasis vectors, the Fourier transforms act via ˆ f ( x α D β ) = ( − | β | D α x β and ˇ f ( x α D β ) = ( − | α | D α x β It should be noted that the Fourier transform ˆ f , unlike the involutivetransposition ˆ J , has periodicity four just as its group-theoretic parenton the Heisenberg clock (Figure 2). It square ˆ f = ˆ J ˇ J = ˇ J ˆ J is theautomorphism A n ( C ) → A n ( C ) induced by x (cid:55)→ − x, ∂ (cid:55)→ − ∂ ; itis denoted by an overbar in [102, §V2b]. Combined with the above-mentioned identification A n ( C ) ∼ = A n ( C ) o with x ↔ ∂ , this yields theso-called principal anti-automorphism [21, §I.2.4] of the Weyl algebra(induced by the inversion map on the group level).It remains to study the interaction between the Weyl algebra A n ( C ) and the twain algebra structure of (cid:83) ( R n ) as well as the Fourier oper-ator (cid:70) : (cid:83) ( R n ) (cid:55)→ (cid:83) ( R n ) . In both cases, the relevant relations followfrom the corresponding action of the Heisenberg group. Thus each D k acts as a derivation on (cid:83) ( R n ) • but as a scalar on (cid:83) ( R n ) (cid:63) while each x j is a scalar for (cid:83) ( R n ) • but a derivation for (cid:83) ( R n ) (cid:63) ; thus the recto/versodistinction of §2.2 transfers from H ( (cid:36) ) to A n ( C ) .While the relations between A n ( C ) and the pointwise structure areclear, those for the convolution follow by the Fourier transform and thewell-known differentiation laws (43) ˆ (cid:70) ( T · s ) = ˆ f ( T ) · ˆ (cid:70) ( s ) , ˇ (cid:70) ( T · s ) = ˇ f ( T ) · ˇ (cid:70) ( s ) for all T ∈ A n ( C ) and s ∈ (cid:83) ( R n ) ; see for example [24, §6], [108, §3.3].As for the Heisenberg action, one may also couch these laws in termsof left and right modules: If the Fourier operators are considered as lin-ear homomorphisms from a left to a right Heisenberg module, we mayadditionally view these as left and right D -modules (with D = A n ( C ) being the Weyl algebra). As for the Heisenberg situation, the rightaction then corresponds to a left action via ˆ J , ˇ J : A n ( C ) → A n ( C ) o forthe Fourier operators ˆ (cid:70) , ˇ (cid:70) .Let us summarize our findings by introducing some tentative ter-minology. By a Weyl action on (cid:83) ( R n ) we mean an action of A n ( C ) satisfying the cross-relations. Since (cid:83) ( R n ) is a twain algebra, the gen-erators D k and x j must interact accordingly (derivations/scalars) withthe multiplications (cid:63) and · ; for plain or slain algebras, these require-ments would be diminished or cancelled, as the case may be. If we have N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 79 a Fourier operator between Heisenberg algbras such that the differenta-tion laws (43) are satisfied, we speak of a compatible
Weyl action. It iseasy to see that similar statements can be made, via the correspond-ing transpose operators, about the twain module (cid:83) (cid:48) ( R n ) of tempereddistributions. Altogether, we have then the following result. Proposition 63.
The Fourier singlets [ (cid:70) : (cid:83) ( R n ) (cid:55)→ (cid:83) ( R n )] as well as [ (cid:70) : (cid:83) (cid:48) ( R n ) (cid:55)→ (cid:83) (cid:48) ( R n )] are endowed with compatible Weyl action. We conjecture that Proposition 63 has a generalization to compatibleWeyl actions of A G ( C ) and A ˆ G ( C ) on the Schwartz-Bruhat functions [ (cid:70) : (cid:83) ( G ) (cid:55)→ (cid:83) ( ˆ G )] and tempered distributions [ (cid:70) : (cid:83) (cid:48) ( G ) (cid:55)→ (cid:83) (cid:48) ( ˆ G )] foran arbitrary Pontryagin duality (cid:36) : ˆ G × G → C . In that case, theFourier transform of the Weyl algebra ˆ f : A G ( C ) → A ˆ G ( C ) would mapthe real character χ : G → ( R , +) to the derivation D χ := iτ ∂ ˆ χ inducedby the one-parameter group ˆ χ ∈ Lie( ˆ G ) , as detailed in Remark 62,while correspondingly mapping D χ to − χ . Remark 64.
Since the Schwartz class (cid:83) ( R n ) and its distribution mod-ule (cid:83) (cid:48) ( R n ) play a prominent role in the theory of Fourier operatorswhile at the same time having the structure of a differential algebrawith derivation ∂ , a short foray into their “integro-differential struc-ture” is certainly not out of place. We shall see, however, that in thiscase we do not have integro-differential algebras [96, 28] or integro-differential modules [99, Lem. 14], not even differential Rota-Baxteralgebras or modules [39, Ex. 3.8c].Let us therefore recall the terminology of [42, §4.1]. Given a dif-ferential algebra ( (cid:70) , ∂ ) , a (linear) section of ∂ is called an antideriva-tive and a (linear) quasi-inverse a quasi-antiderivative. Note that an-tiderivatives are a special case of quasi-antiderivatives. What we shallneed in the sequel is their converse: (linear) retractions of (injectivebut non-surjective!) derivations. Let us refer to these, tentatively, as retro-antiderivatives .For the sake of simpler notation, we state only the ordinary case withderivations ∂ : (cid:83) ( R ) → (cid:83) ( R ) and ∂ : (cid:83) (cid:48) ( R ) → (cid:83) (cid:48) ( R ) and (quasi)inversesto be given. It is then straightforward to set up the corresponding par-tial operators ∂ x , ∂ y , . . . with their (quasi)inverses as in [94].Recall that the space of bump functions (cid:68) ( R ) , meaning smooth func-tions of compact support, is a (nonunital) differential algebra that doesnot admit antiderivatives since its derivation is indeed non-surjective:Writing (cid:117) : (cid:68) ( R ) → C for the definite integral (cid:117) s := (cid:114) ∞−∞ s ( ξ ) dξ , onehas s ∈ im( ∂ ) ⇔ (cid:117) s = 0 according to [106, Lem. 2.5.1]. This conditionalso characterizes the image of ∂ : (cid:83) ( R ) → (cid:83) ( R ) . Indeed, if s = S (cid:48) for ∗ some S ∈ (cid:83) ( R ) , we have S ( −∞ ) = 0 and thus(44) S ( x ) = (cid:114) x −∞ s ( ξ ) dξ, which for x → + ∞ implies (cid:117) s = 0 . Conversely, assuming the lattercondition on s implies for S defined by (44) that for any n > wehave | S ( x ) | ≤ C | x | − n +1 where C = C n is chosen so that | s ( ξ ) | ≤ C | ξ | − n ,using the fact that s ∈ (cid:83) ( R ) . This implies S = O ( | x | − k as x → −∞ forany k > while S = O ( | x | ) = O (1) follows already from S ( −∞ ) = 0 by the definition (44). For the asymptotics as x → + ∞ , we use (cid:117) s = 0 to write the integral as S ( x ) = (cid:114) ∞ x s ( ξ ) dξ , and as before we obtain S ( x ) = O ( x − k ) for any k > , and this time the case k = 0 followsfrom (cid:117) s = 0 .We can set up a retro-antiderivative (cid:30) (cid:114) both on (cid:68) ( R ) and on (cid:83) ( R ) .Following [108, Exc. 2.6.17] and [106, §2.5], we choose a bump function s ∈ (cid:68) ( R ) ⊆ (cid:83) ( R ) with (cid:117) s = 1 , say s ( x ) := c exp( x − ) H (1 − | x | ) ,where c is a normalization constant [106, Exc. 2.1.1]. Then we define (cid:30) (cid:114) on (cid:68) ( R ) and (cid:83) ( R ) by applying the indefinite integral operator s (cid:55)→ S of (44) to s − ( (cid:117) s ) s ker (cid:117) ∈ im ∂ instead of s . Thus we set(45) (cid:30) (cid:114) s := (cid:114) x −∞ (cid:16) s ( ξ ) − ( (cid:117) s ) s ( ξ ) (cid:17) dξ, and it is easy to see that (cid:30) (cid:114) ∂ = 1 so that (cid:30) (cid:114) is surjective. But (cid:30) (cid:114) isnot injective since clearly ker (cid:30) (cid:114) = [ s ] . We obtain the direct decom-positions im ∂ (cid:117) ker (cid:30) (cid:114) = (cid:68) ( R ) and im ∂ (cid:117) ker (cid:30) (cid:114) = (cid:83) ( R ) associatedwith the projector s (cid:55)→ s − ( (cid:117) s ) s . As we have seen, these spacesare im ∂ = ker (cid:117) and ker (cid:30) (cid:114) = [ s ] . The choice of s picks out a com-plement of the deficient image of the derivative, which engenders thekernel of the retro-antiderivative (cid:30) (cid:114) .It is not to be expected that (cid:30) (cid:114) be a Rota-Baxter operator, in thesense of satisfying ( (cid:30) (cid:114) s )( (cid:30) (cid:114) s ) = (cid:30) (cid:114) s (cid:30) (cid:114) s + (cid:30) (cid:114) s (cid:30) (cid:114) s . Indeed, the correctionterm ( (cid:117) s ) (cid:30) (cid:114) s (cid:30) (cid:114) s + ( (cid:117) s ) (cid:30) (cid:114) s (cid:30) (cid:114) s is required (on the left-hand side), soweight terms [42, Def. 2.1b] are of no avail in this case. Instances witha nonzero correction are easy to come by: For example, taking s = s and s = s (cid:48) leads to the correction ε = (cid:114) s with | ε (0) | > . .Let us now turn to the distribution spaces (cid:68) (cid:48) ( R ) and (cid:83) (cid:48) ( R ) , whereone has a well-known antiderivative [106, (5.6)], namely the transposeof − (cid:30) (cid:114) . By abuse of notation, we shall also denote it by (cid:30) (cid:114) . Since thederivation ∂ on (cid:68) (cid:48) ( R ) and (cid:83) (cid:48) ( R ) are also defined as the tranpose of thederivation − ∂ on the corresponding primal spaces, it is clear that (cid:30) (cid:114) isan antiderivative on the distribution spaces. Moreover, we see that ∂ is surjective and (cid:30) (cid:114) is injective for distributions, just as in the familiardifferential Rota-Baxter and integro-differential settings [39, Ex. 3.8cd]. N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 81
Indeed, the kernel of ∂ is easily seen [106, §2.5] to be the constantdistributions R ⊂ (cid:83) (cid:48) ( R ) ⊂ (cid:68) (cid:48) ( R ) , and the image of (cid:30) (cid:114) the distributionsannihilating s . In other words, the initialization − (cid:30) (cid:114) ∂ is the projectorthat maps a distribution ˜ s to the constant distribution ˜ s ( s ) ∈ R .In view of the results for (cid:68) ( R ) and (cid:83) ( R ) , one will anticipate that (cid:30) (cid:114) on (cid:68) (cid:48) ( R ) and (cid:83) (cid:48) ( R ) does not satsify the Rota-Baxter axiom in theform ( (cid:30) (cid:114) s )( (cid:30) (cid:114) ˜ s ) = (cid:30) (cid:114) ˜ s (cid:30) (cid:114) s + (cid:30) (cid:114) s (cid:30) (cid:114) ˜ s , where s ranges over (bump or Schwartz)functions and ˜ s over (general or tempered) distributions, with the con-venient notation (cid:30) (cid:114) accordingly overloaded. This axiom would make thecorresponding distribution spaces into differential Rota-Baxter mod-ules [39, Ex. 3.8c], but again we have (on the left-hand side) a correc-tion term ( (cid:117) s ) s (cid:30) (cid:114) ˜ s + ˜ s ( (cid:30) (cid:114) s (cid:30) (cid:114) s ) , which can easily be worked out fromthe corresponding primal correction.Before ending this chapter, a few words seem to be in order about the Laplace transform , where the all-important differentiation laws abovecontinue to hold. In fact, they could be said to really come into theirown since analyticity enters the picture. Since the relevant ideas havebeen developed in the general setting of Pontryagin duality, we shallnow leave the specific setting of Schwartz class on R n , returning to thegeneral Schwartz-Bruhat space (cid:83) ( G ) on an LCA group. Remark 65.
Recall the complex characters discussed in Remark 62.For an LCA group G under Pontryagin duality (cid:36) : ˆ G × G → T , theseare continuous homomorphisms ζ : G → C × = R + × T . If the spaceof characters G → ( R , +) is denoted by G as in [66, Def. 2], everycomplex character has a unique polar decomposition ζ = e ρ ξ for ρ ∈ G and ξ ∈ ˆ G .We set Γ := G × ˆ G and define the extended Pontryagin duality ˜ (cid:36) : Γ × G → C × by (cid:104) ρ, ξ | x (cid:105) ˜ (cid:36) := e ρ ( x ) (cid:104) ξ | x (cid:105) (cid:36) . It is indeed a duality overthe torus C × as can check easily. Note also that T G (cid:111) ˆ G ≤ C × G (cid:111) Γ ,meaning H ( (cid:36) ) ≤ H ( ˜ (cid:36) ) , so Heisenberg actions over ˜ (cid:36) restrict to thoseover (cid:36) . The Laplace transformation of [66] is expected to carry overto the setting of Heisenberg algebras in the following sense.Writing ˙ L ( G ) ⊆ L ( G ) for strongly L functions [66, Def. 2] definin-ing C ω ( ˜ G ) as the functions analytic in the sense of [66, Def. 9] on U × ˆ G for a convex open zero neighborhood U ⊆ G , we obtain a regularFourier doublet [ (cid:76) : ˙ L ( G ) (cid:55)→ C ω ( ˜ G )] as per Theorem 7 of [66]. Tobe precise, the “functions” of C ω ( ˜ G ) should be understood as suitablydefined partial germs (direct limit indexed over subsets U × ˆ G ⊂ Γ ofthe form specified above), with the L Laplace transformation [69], [66] ∗ given by (cid:76) s ( ρ, ξ ) = (cid:70) ( e ρ s )( ξ ) = (cid:90) G e ρ ( x ) (cid:104) ξ | x (cid:105) (cid:36) s ( x ) dx in terms of the usual L Fourier transform (Proposition 54). It shouldbe clear, however, that the
Laplace doublet [ (cid:76) : ˙ L ( G ) (cid:55)→ C ω ( ˜ G )] isdefined over (cid:36) rather than ˜ (cid:36) since the action of G ≤ H ( ˜ (cid:36) ) is onlylocal.The notion of differential operator (39) sketched above is also ap-plicable to functions (locally) defined on Γ = G × ˆ G . Indeed, a one-parameter subgroup is given by R × Γ → Γ , t · ( ρ, ξ ) = ( ρ + tρ, ξ + tξ ) foran arbitrary ( ρ, ξ ) ∈ Γ . In contrast to [66, (1)] we write ˆ G additively,identifying one-parameter subgroups on ˆ G as in Remark 62 with realcharacters ξ so that ( ξ + tξ )( x ) = e iτξ ( x ) t ξ ( x ) for x ∈ G and t ∈ R .According to [66, (1)], the partial derivative of a spectrum σ ∈ C ω ( ˜ G ) is then defined as ∂σ∂ζ ( ζ ) = lim t → σ ( ρ + tρ, ξ + tξ ) t at the point ζ = ( ρ, ξ ) ∈ ˜ G ⊆ Γ in the direction ζ = ( ρ, ξ ) . Writing thelatter ζ = ρ + iξ induces a complex structure on Γ as per [66, (2)], sothat the notion of analyticity means C -linearity along with certain L requirements.The upshot is that the differentiation law ∂ ζ (cid:76) = (cid:76) ζ holds on ˙ L ( G ) ,as reported in [66, Thm. 5]. (But note a typo: the function on theleft-hand side there misses the Laplace sign.) Here the natural actionof ζ = ρ + iξ ∈ Γ on ˙ L ( G ) is componentwise, endowing C ω ( ˜ G ) withthe structure of a C [Γ] -module. Of course, also ˙ L ( G ) is a C [Γ] -modulevia ζ · s = ∂ ζ s , and then (cid:76) : ˙ L ( G ) → C ω ( ˜ G ) is a C [Γ] -linear homo-morphism.There are two essential instances of the Laplace doublet (each withits obvious multidimensional generalizations): • Choosing G = Z leads to Laurent series [66, Ex. 1]. One ob-tains ˆ G = T and G = R . The Laplace transform is (cid:76) s ( ρ, ξ ) = ∞ (cid:88) n = −∞ s n ρ n e iτnξ , where we may read ρ ∈ R + as radius and ξ ∈ T ∼ = R / Z as angle measure. The region of convergence (ROC) is someannulus ˜ G ⊆ U × R + with T ⊆ U . N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 83 • On the other hand, taking G = R yields the classical bilat-eral Laplace transformation [66, Ex. 2]. Its ROC is a verticalstrip ˜ G ⊆ U × R with horizontal span U ⊆ R .For the univariate bilateral Laplace transformation (also known as“Fourier-Laplace transformation” when applied to distributions), onehas the so-called Paley-Wiener theorems [108, 7.2.1-4] characterizingthe compactly supported elements of L ( R ) , C ∞ ( R ) , (cid:68) (cid:48) ( R ) , L ( R + ) via growth rates of their transforms.Laplace transforms are of course ubiquitous in diverse applications.The bilateral version mentioned above is related to the more com-mon unilateral Laplace transform (cid:76) + by (cid:76) = (cid:76) + + (cid:76) − , where onesets (cid:76) − := (cid:80)(cid:76) + (cid:80) with the reversal operator (cid:80) of Theorem 37. Con-versely, one obtains the unilateral transformation from the bilateralone via (cid:76) ± = (cid:76) h ± where h ± is (the multiplication operator associatedwith) the characteristic function of the half-line R ± , also known as the(ascending/descending) Heaviside function [99, §3]. The presence ofthe Heavisides leads to Dirac terms upon differentation, which are inturn responsible for the characteristic initial values in differential re-lations such as (cid:76) ∂ ζ = ζ (cid:76) − ev , where ev means evaluation at as a right-handed limit. For solving initial value problems, the evalua-tion term is crucial since it allows to incorporate the given initial data.Another advantage of the unilateral transformation over its bilateralsibling is that its ROC is usually larger. (Intuitively speaking, the bi-lateral transformation converges only on a strip but the unilateral oneon a half-plane.)Note that the differentation law just mentioned for the unilateralLaplace transformation is the converse of the one mentioned abovefor the bilateral transformation. It would be worthwhile to work outsuitable extended function spaces based on the tempered distributionspaces (cid:83) (cid:48) ( G ) and (cid:83) (cid:48) ( ˆ G ) such that both differential laws hold for thegeneralized unilateral Laplace transform (without evalation terms), aswith the compatible Weyl action of Proposition 63. We expect thatsuch function spaces would enjoy a compatible action of an extendedWeyl algebra , which is “thickened” to contain complex multipliers ζ = ρ + iξ on signals and the corresponding differential operators ∂ ζ onspectra.Physically speaking, this extension from Fourier to Laplace transfor-mations may be interpreted as follows: The Fourier approach is basedon analyzing/synthesizing signals into/from spectra that are essentiallypure oscillations. The Laplace approach generalizes pure oscillations to ∗ damped oscillations , with the possibility to control the damping factoras an additional parameter (via real characters).From the algebraic point of view, we may summarize the role of theLaplace transform as follows: As opposed to the Fourier transform, itappears to be less amenable to a global description such as we havefor the Fourier transform (where we have many classical examples ofFourier doublets in various gradations). For the time being, we tend tosee it more as a local tool that facilitates the close-up study of a fixed function by the tools of complex analysis.Another algebraic aspect of the Laplace transform would be inter-esting to pursue further in the present context: In the classical case of G = Γ = R , the functions supported on R + form an important subal-gebra (cid:0) L ( R + ) , (cid:63) (cid:1) that is known to be an integral domain (the so-calledTitchmarsh Theorem). Its fraction field is the main object of study forthe Mikusiński calculus , a kind of algebraic formulation of the Laplacetransform [74], [120]. In particular, the Heaviside function h ≡ { } of [120, §2] has as its reciprocal s a kind of (bijective!) differentiationoperator. One may then adjoin further hyperfunctions such as e −√ s in [120, §27], which is an algebraic version of the heat propagator.It would be very intersting to investigate the relationship betweenthis calculus and localized Fourier doublets. For example, it is clearthat the Gaussian D -module (87) developed in §4.1 below is an integraldomain and so could be localized. For getting something like the hy-perfunction s , one might start with h (cid:83) ( R ) := { h t f | f ∈ (cid:83) ( R ) , t ∈ R } ,where h t := t • h are the translated Heaviside functions. By a versionof the Titchmarsh Theorem, h (cid:83) ( R ) ⊂ (cid:0) L ( R ) , (cid:63) (cid:1) is an integral domainand thus has a fraction field that might profitably be compared withthe Mikusiński field. (cid:125) Constructive Fourier Analysis via Schwartz Functions
A Minimal Subalgebra of the Schwartz Class.
We shall nowfocus on the Schwartz singlet (cid:83) (cid:104) R n | R n (cid:105) corresponding to the standardvector duality of Example 9 since this is the most important case forapplications of Fourier analysis to partial differential equations. Wewill define two subsinglets ¯ (cid:71) (cid:104) R | R (cid:105) ≤ (cid:83) ( R n ) , continuing at first withthe base field R . This treatment is relatively constructive , assuming anoracle for computations in R . In the subsequent §4.2, we shall buildup a constructive subfield Q π of R that will allow us to build up therestricted Fourier singlet ¯ (cid:71) (cid:104) Q | Q (cid:105) with the constructive base field Q π so as to allow a purely algebraic and algorithmic description in §3.5. N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 85
For simplicity, we shall take n = 1 . As we have seen in (25), thisis no essential restriction since the multivariate case can be reducedto the univariate one. In fact, the classical Fourier integral (26) ex-hibits this reduction, which works the same in the classical twain singlet L / (cid:104) R n | R n (cid:105) , and the latter contains the Schwartz singlet (cid:83) (cid:104) R n | R n (cid:105) byProposition 58.Our intention is to set up a subalgebra of (cid:83) ( R ) that is as simple aspossible. Since we need functions of rapid decay, polynomials will notdo. The simplest choice that comes to mind is given in terms of the Gaussian normal distribution . Fixing mean µ and variance (cid:14) τ ρ , thecorresponding probability density is ρ / g µ,ρ with g µ,ρ ( x ) := e − πρ ( x − µ ) .We prefer to avoid normalization factors that involve √ π , so we shallonly work with the unnormalized Gaussian distribution functions g µ,ρ ,which we shall briefly call Gaussians . Their C -linear span yields theimportant algebra(46) (cid:71) ( R ) := [ g µ,ρ | ( µ, ρ ) ∈ R × R ≥ ] C ≤ C ∞ ( R ) under pointwise multiplication. Indeed, we have the explicit productlaw g µ ,ρ g µ ,ρ = c g µ,ρ for ρ ρ > , where the Gaussian parametersare ρ := ρ + ρ and µ := ( ρ µ + ρ µ ) /ρ , and where c := e − πρ ( µ − µ ) with ρ := ρ ρ /ρ is a (relative) normalization constant. Of course,the case ρ = ρ = 0 is trivial since g µ, = 1 . In fact, this is the onlyelement that is not contained in the Schwartz class (cid:83) ( R ) since doesnot have rapid decay (well—no decay at all). Hence we shall prefer towork with the (nonunital!) subalgebra(47) (cid:71) ( R ) := [ g µ,ρ | ( µ, ρ ) ∈ R × R > ] C ≤ (cid:71) ( R ) , whose unitalization is of course (cid:71) ( R ) = R ⊕ (cid:71) ( R ) since g µ, = 1 for all µ ∈ R . As mentioned before, we have now (cid:71) ( R ) ≤ (cid:83) ( R ) asa subalgebra under the pointwise product; we call this the Gaussianalgebra .For establishing C -linear independence of the Gaussians, one mayusefully appeal to asymptotic notions for x → + ∞ ; of course onecould equally use x → −∞ . In the sequel, all limits and germs arefor x → + ∞ . For any such functions f, g : R → C , we recall that f is called negligible with respect to g when lim f /g = 0 ; this is eitherwritten as a relation f Î g following Hardy or with Landau’s little ohnotation f = o ( g ) .While the collection of all germs is a ring, the smooth ones clearlyform a differential ring R ∞ . It is often expedient, though, to workwith differential subfields of R ∞ , known as Hardy fields [10, §1]: Theycome with a canonical total order, so are ordered fields; moreover, their ∗ germs all have definite limits on the extended real line R ∪ {±∞} . Forexample, the Hardy field of logarithmic-exponential functions [10, §1],[41, Ex. 1] is certainly large enough for our modest purposes.Given any Hardy field (cid:70) , one may want to isolate an asymptoticscale (cid:69) within (cid:70) . Similar to [41, §2.5], we define this to be a multiplica-tive subgroup of (cid:70) + := { f ∈ (cid:70) | f > } totally ordered under Î . Thenthe subfield C ( (cid:69) ) generated by (cid:69) is endowed with a valuation, inducedfrom the natural valuation [10, V1–3] of (cid:70) . Its valuation ring consistsof all (germs of) functions in C ( (cid:69) ) that remain finite for x → + ∞ .Following [34, §III.2], we consider specifically the asymptotic scale (cid:69) consisting of the unity germ and all germs of the form f ( x ) = x α log β xe P ( x ) with α, β ∈ R and P ( x ) ∈ R [ R > ] . The representation of non-unitygerms is unique if we stipulate that α, β (cid:54) = 0 . Furthermore, we agreeto write the exponents in descending order P ( x ) = a x γ + · · · + a k x γ k ,so that γ > · · · > γ k > and a , . . . , a k ∈ R . It is clear that allsuch germs are positive, and they form a group under multiplication(and one may also take powers with arbitrary real exponents). Theircrucial property for asymptotic investigations, however, is that anysuch germ f ( x ) (cid:54) = 1 either tends to or to ∞ . In detail, we have(48) x α (log x ) β e P ( x ) Ï x ˜ α (log x ) ˜ β e ˜ P ( x ) iff ( P ( x ) , α, β ) > ( ˜ P ( x ) , ˜ α, ˜ β ) under the lexicographic order on R [ R > ] × R × R . Here the monoidring R [ R > ] is itself given the lexicographic order a x γ + · · · + a k x γ k Ï ˜ a x γ + · · · + ˜ a k x γ k iff ( a , · · · , a k ) > (˜ a , · · · , ˜ a k ) , where one pads P ( x ) and ˜ P ( x ) with zero coefficients a j so that they exhibit the same expo-nent sequence γ > · · · > γ k > .The elements of (cid:69) are all linearly independent over C , so they forma C -basis for the subspace of germs generated by (cid:69) . For seeing this,assume(49) c f + · · · + c n f n = 0 is any linear relation among distinct germs f , . . . , f n ∈ (cid:69) with coef-ficients c , . . . , c n ∈ C × and length n > . Without loss of generality,we may assume that f Ï · · · Ï f n . Multiplying (49) with c − f − , weobtain(50) c ˜ f + · · · + ˜ c n ˜ f n = 0 with ˜ c i := c − c i and ˜ f i := f − f i for i = 2 , . . . , n . Note that we haveagain descending asmptotic growth Ï ˜ f Ï · · · Ï ˜ f n . By transitivity N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 87 of Ï , this implies that ˜ f , . . . , ˜ f n → so that (50) yields upon taking the limit x → + ∞ .It is easy to see that the induced valuation on C ( (cid:69) ) is essentiallygiven by the exponents in (48), except for the conventional sign (sothe valuation measures decay rather than growth at infinity). In otherwords, we may define (51) ν ( x α log β xe P ( x ) ) = (cid:0) − P ( x ) /π, − α, − β (cid:1) , for the scale functions of (cid:69) . This extends to C -linear combinationsof scale functions f , f ∈ (cid:69) via ν ( c f + c f ) = min { ν ( f ) , ν ( f ) } bythe ultrametric inequality. In this way, we have defined the valuationon the subalgebra C [ (cid:69) ] ⊂ C ( (cid:69) ) , which determines the general valua-tion ν : C ( (cid:69) ) → R [ R > ] × R × R by ν ( f /f ) = ν ( f ) − ν ( f ) . Clearly, C [ (cid:69) ] is the valuation ring corresponding to ν , hence in particular anintegral domain. It is moreover the group ring over the ordered abeliangroup (cid:71) , so the valuation ν : C [ (cid:69) ] → R [ R > ] is essentially a specialcase of [35, Exc. 11.4].For our purposes, we shall only need the exponentials with standardpolynomials in the exponent. Since we have to ensure that the constantfunction is the only germ not tending to or to ∞ , we shall use theadditive group R [ x ] + of polynomials with positive degree to introduce (cid:69) := { e P ( x ) | P ( x ) ∈ R [ x ] + } ⊂ (cid:69) which is easily seen to form an asymptotic subscale with correspondingfield C ( (cid:69) ) and valuation ring C [ (cid:69) ] . Again, the exponentials of (cid:69) are then linearly independent over C , so the C -linear span of (cid:69) is(isomorphic to) the group algebra C [ R [ x ] + ] , a rare case of iteratedmonoid algebra. The valuation restricts to ν : C ( (cid:69) ) → R [ x ] + .Focusing on the subalgebra (cid:71) ( R ) ⊂ C [ R [ x ] + ] , the valuation restrictsfurther to a semigroup epimorphism ν : C × (cid:71) • (cid:16) R > x + R x , with C × (cid:71) • := { cg | c ∈ C × , g ∈ (cid:71) • } the multiplicative subsemigroupof (cid:71) ( R ) generated by the Gaussians (cid:71) • := { g µ,ρ | ( µ, ρ ) ∈ R × R > } . Indetail, we have ν ( cg µ,ρ ) = ρx − ρµx , and its kernel by the relation ∼ with cg µ,ρ ∼ ˜ cg ˜ µ, ˜ ρ iff ( µ, ρ ) = (˜ µ, ˜ ρ ) , which implies C × (cid:71) • / ∼ ∼ = (cid:71) • .Identifying (cid:71) • with R × R > via g µ,ρ ↔ ( µ, ρ ) , the quotient monoidstructure is given by(52) ( µ , ρ ) · ( µ , ρ ) = ( µ + ρ µ , ρ + ρ ) , where µ + ρ µ := ( µ ρ + µ ρ ) / ( ρ + ρ ) denotes the ρ -weightedarithmetic mean of µ and µ for the weight vector ρ := ( ρ , ρ ) . By The extra factor π − for the polynomial exponent is only convenience for laterpurposes. ∗ the first isomorphism theorem (for semigroups), the semigroup (cid:71) • isjust the additive semigroup R > x + R x ∼ = R > ⊕ R in disguise. Inthe sequel, we shall refer to (cid:71) • as the pointwise Gaussian semigroup .Let us now reconstruct the full algebra structure on (cid:71) ( R ) . Sincethis is just the unitalization of (cid:71) ( R ) , it suffices to build up the structureof the latter. We shall use the strategy of group homology [46, § VI]for this purpose, but adopted to our present setting. Hence assume(53) (cid:47) (cid:47) C (cid:31) (cid:127) (cid:47) (cid:47) H π (cid:47) (cid:47) G s (cid:106) (cid:106) (cid:47) (cid:47) is an exact sequence of semigroups in the sense that ker( π ) is the con-gruence on H given by h ∼ h (cid:48) iff h = ch (cid:48) for a unique c ∈ C . Then thequotient h/sπ ( h ) ∈ C is well-defined for any h ∈ H . While here weshall only need the fully abelian setting, it is natural to allow H to benonabelian as long we retain commutativity on the orbits (see belowfor the precise set of axioms). In this way, we include central extensionsof abelian groups such as the ones used for studying Heisenberg groupsin § (cid:74) ?? (cid:75) .Since quotients are thus well-defined, we may form the cocycle (or“factor set”)(54) ψ : G × G → C, ( g, g (cid:48) ) (cid:55)→ s ( g ) s ( g (cid:48) ) s ( gg (cid:48) ) , just as in the group case [46, Exc. 10.1]. It should be emphasized,however, that C is not required to be embedded in H ; we only requirea compatible semigroup action · : C × H → H , where compatibility heremeans that · is a homomorphism of semigroups (with C × H being thedirect product). This is what the wavy arrow in (53) is supposed toconvey. Let us note that the action · : C × H → H is automatically free in the sense that c (cid:54) = c (cid:48) implies c · h (cid:54) = c (cid:48) · h for all h ∈ H .Equivalently, we can also stipulate freeness while giving up uniquenessin the exactness requirement. Altogether, we have imposed on (53) the axioms c · c (cid:48) · h = cc (cid:48) · h c (cid:54) = c (cid:48) ⇒ c · h (cid:54) = c (cid:48) · h ( c · h )( c (cid:48) · h (cid:48) ) = cc (cid:48) · hh (cid:48) h ∼ h (cid:48) ⇔ ∃ c c · h = h (cid:48) h ∼ h (cid:48) ⇒ hh (cid:48) = h (cid:48) h C, G abelianof action, freeness, compatibility, exactness, orbit commutativity, andabelian flanks. It is easy to check the calculation rules h h (cid:48) h h (cid:48) = h h h (cid:48) h (cid:48) and c · h h (cid:48) = c h h (cid:48) for c ∈ C and h ∼ h (cid:48) , h ∼ h (cid:48) . Moreover, we have thecancellation rules hk kh (cid:48) = hh (cid:48) as well as hk kh (cid:48) = hh (cid:48) for h, h (cid:48) ∼ k . While we N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 89 have not required any of the three semigroups to be monoids or groups,it turns out that C is in fact a group. Lemma 66.
Given an exact sequence as in (53) , the semigroup C isa group whose identity element acts trivially.Proof. Choose any h ∈ H . Then we have hh · h = h by definition andtherefore c hh · h = c · hh · h = c · h for any c ∈ C . This yields hh c = c byfreeness. Since c ∈ C was arbitrary, hh is seen to be an identity elementfor C . But such an element is unique in any semigroup, so we mayunambiguously write hh . Thus C is a monid. It is in fact a groupsince every element c ∈ C can be written as hh (cid:48) for suitable h, h (cid:48) ∈ H , taking for example h := c · h (cid:48) for arbitrary h (cid:48) . Clearly, such anelement has h (cid:48) h for its inverse by the calculation rules mentioned above.Moroever, if h ∈ H is arbitrary, we have · h = hh · h = h so that actstrivially as claimed. (cid:3) With the help of Lemma 66, we can derive two more useful calcu-lation rules : As usual, we have the equality condition for fractionssaying that h h (cid:48) = h h (cid:48) iff h h (cid:48) = h (cid:48) h for h ∼ h (cid:48) and h ∼ h (cid:48) , and wehave the mixed associativity law ( c · h ) h (cid:48) = c · hh (cid:48) for all h, h (cid:48) ∈ H and c ∈ C . The latter follows immediately from the compatibil-ity axiom by substituting c (cid:48) = 1 . For showing the former, assumefirst the equality of fractions and set c := h /h (cid:48) = h /h (cid:48) to ob-tain h h (cid:48) = ( c · h (cid:48) )( c − · h ) = h (cid:48) h . Conversely, assuming this conditionwe get h = h h (cid:48) h (cid:48) h · h = h h (cid:48) · h h (cid:48) · h = h h (cid:48) · h (cid:48) , which implies the desiredequality of fractions by the uniqueness of quotients.Even though H is only a (nonabelian) semigroup, our axioms on (53)allows us to define the commutator [ , ] : H × H → C via [ h, k ] := hkkh .Note that this quotient is well-defined since we have hk ∼ kh by thecommutativity of G . Now assume h ∼ h (cid:48) and k ∼ k (cid:48) . We claimthat [ h (cid:48) , k (cid:48) ] = [ h, k ] since this is equivalent to h (cid:48) k (cid:48) kh = k (cid:48) h (cid:48) hk , which isin turn equivalent to cd · hkkh = cd khhk with c := h (cid:48) /h and d := k (cid:48) /k .But the latter equality follows immediately from orbit commutativitysince hk = kh . In analogy to the group case in Definition (cid:74) ?? (cid:75) , wecan thus introduce the commutator form ω : G × G → C by ω ( g, g (cid:48) ) =[ s ( g ) , s ( g (cid:48) )] , where the choice of the (set-theoretic) section s : G → H is immaterial by what we have just shown. Moreover, ω is clearyantisymmetric and it is linear since even [ h h , h ] = [ h , h ] [ h , h ] forall h , h , h ∈ H . Indeed, the latter is true iff(55) h h hhh h = h hhh h hhh , ∗ which we establish now. Writing c := h hhh and d := h hhh for the right-hand quotients, we get cd · hh h = ( c · hh )( d · h ) = h h ( d · h ) = h h h ;now (55) follows as usual by the uniqueness of quotients.Let us now state how the cocycle (54) encodes the structure of thesemigroup H and its extension as algebra. Lemma 67.
For an exact sequence (53) with fixed section s : G → H ,the cocycle (54) induces a semigroup C × ψ G isomorphic to H .If C = ( K × , · ) for a commutative unital ring K , the product of H hasa unique K -linear extension to K ( G ) ∼ = K ψ [ G ] .Proof. As one checks immediately, one obtains on the level of sets abijection(56) H ∼ = C × G : (cid:40) h (cid:55)→ (cid:0) h/sπ ( h ) , π ( h ) (cid:1) ,c s ( g ) ← (cid:91) ( c, g ) . which becomes an isomorphism of semigroups by transporting the op-eration from H to C × G . It is easy to see that the new operation · ψ isgiven by ( c, g ) · ψ ( c (cid:48) , g (cid:48) ) = (cid:0) ψ ( g, g (cid:48) ) cc (cid:48) , gg (cid:48) (cid:1) , using (56) and the compati-ble action · : C × H → H . We write C × ψ G for the resulting semigroup.Note that every element of h ∈ H can be written uniquely as h = c s ( g ) with c ∈ C and g ∈ G , where h ↔ ( c, g ) in (56). Identifying G via s asa subset of H , we shall write this simply as h = cg .Now assume C = K × for a commutative unital ring K . In that case,the unitarizations Kg of the congruence classes [ g ] ∼ = Cg of H/ ∼ arefree K -modules and direct components of K ( G ) ∼ = (cid:76) g ∈ G Kg . Using thepartition of H into its congruence classes, there is an injective map H = (cid:93) g ∈ G [ g ] ∼ (cid:44) −→ K ( G ) sending cg ∈ H to ce g , where ( e g | g ∈ G ) is the basis of K ( G ) . Identi-fying H as a subset of K ( G ) , its product induces for any g, g (cid:48) ∈ G theunique K -linear map(57) m g,g (cid:48) : Ke g ⊗ Ke g (cid:48) → Ke gg (cid:48) , given by ψ ( g, g (cid:48) ) ∈ C with respect to the bases { e g ⊗ e g (cid:48) } and { e gg (cid:48) } .Since K ( G ) ⊗ K ( G ) is the direct sum of all Ke g ⊗ Ke g (cid:48) , the maps m g,g (cid:48) combine to a bilinear product map · : K ( G ) × K ( G ) → K ( G ) . Endowedwith this product, the algebra ( K ( G ) , + , · ) must now be shown isomor-phic to the twisted semigroup algebra K ψ [ G ] . Writing the generatorsof the latter as θ g ( g ∈ G ) , the map e g (cid:55)→ θ a is obviously a K -linear N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 91 isomorphism
Φ : K ( G ) ∼ −→ K ψ [ G ] , so it remains to prove that Φ respectsmultiplication. But this is immediate from Φ( e g e g (cid:48) ) = Φ( ψ ( g, g (cid:48) ) e gg (cid:48) ) = ψ ( g, g (cid:48) ) Φ( e gg (cid:48) )= ψ ( g, g (cid:48) ) θ gg (cid:48) = θ g θ g (cid:48) = Φ( e g ) Φ( e g (cid:48) ) , using (57) and the definition of multiplication in K ψ [ G ] . (cid:3) We define the category of exact sequences of semigroups (53)
SES sg by taking as morphism ( χ, Φ , ϕ ) : E → E those χ ∈ Hom( C , C ) , Φ ∈ Hom( H , H ) , ϕ ∈ Hom( G , G ) where the right square commutesand where Φ is equivariant over χ . In detail, we require π ◦ Φ = ϕ ◦ π and Φ( c · h ) = χ ( c ) · Φ( h ) for c ∈ C and h ∈ H . It is easy to see that thisimplies χ ( hh (cid:48) ) = Φ( h )Φ( h (cid:48) ) for all h ∼ h (cid:48) ∈ H , hence in particular χ (1) = 1 .For Heisenberg groups, one can characterize the morphisms in terms ofcoboundaries; see Proposition (cid:74) ?? (cid:75) . Here we have an analogous resultfor the semigroup case. Proposition 68.
Let E be the exact sequence (53) and E (cid:48) its primedversion with cocycles ψ, ψ (cid:48) from sections s : G → H and s (cid:48) : G (cid:48) → H (cid:48) ,respectively. Then Hom(
E, E (cid:48) ) = { ( χ, Φ , ϕ ) : E → E (cid:48) } is in bijectivecorrespondence with { ( χ, ϕ, ζ ) | χ ∈ Hom(
C, C (cid:48) ) ∧ ϕ ∈ Hom(
G, G (cid:48) ) ∧ ζ ∈ C ( G, C (cid:48) ) ∧ d ( ζ ) = χ ∗ ψϕ ∗ ψ (cid:48) } such that ( χ, Φ , ϕ ) corresponds to ( χ, ϕ, ζ ) with ζ ( g ) = Φ( sg ) /s (cid:48) ϕ ( g ) and Φ( h ) = ζ ( πh ) χ ( h/sπh ) · s (cid:48) ϕ ( πh ) .Proof. For fixed χ ∈ Hom(
C, C (cid:48) ) and ϕ ∈ Hom(
G, G (cid:48) ) , we will showthat the given relation Φ ↔ ζ sets up a bijective correspondence be-tween arbitrary functions ζ : G → C (cid:48) and those functions Φ : H → H (cid:48) that satisfy π ◦ Φ = ϕ ◦ π and Φ( c · h ) = χ ( c ) · Φ( h ) while not neces-sarily being a homomorphism (so we neither require ( χ, Φ , ϕ ) ∈ SES sg nor d ( ζ ) = χ ∗ ψϕ ∗ ψ (cid:48) ).First of all, it is easy to see that, for given ζ : G → C (cid:48) , the inducedfunction Φ does satisfy π ◦ Φ = ϕ ◦ π as well as Φ( c · h ) = χ ( c ) · Φ( h ) .Let us now check bijectivity of the relation Φ ↔ ζ . Defining ζ for agiven Φ , we obtain ˜Φ( h ) = ζ ( πh ) χ ( h/sπh ) · s (cid:48) ϕ ( πh ) = χ ( h/sπh ) Φ( sπh ) s (cid:48) ϕ ( πh ) · s (cid:48) ϕ ( πh )= χ ( h/sπh ) · Φ( sπh ) = Φ( h )Φ( sπh ) · Φ( sπh ) = Φ( h ); ∗ starting from a given ζ , we get the corresponding Φ and then ˜ ζ ( g ) = Φ( sg ) s (cid:48) ϕ ( g ) = ζ ( πsg ) χ ( sg/sπsg ) · s (cid:48) ϕ ( πsg ) s (cid:48) ϕ ( g ) = ζ ( g ) χ ( sg/sg ) · s (cid:48) ϕ ( g ) s (cid:48) ϕ ( g ) = ζ ( g ) . Thus we have established bijectivity. Next we compute Φ( h ¯ h ) = ζ ( g ¯ g ) χ (cid:0) h ¯ h/s ( g ¯ g ) (cid:1) · s (cid:48) ( ϕg ϕ ¯ g ) , Φ( h ) Φ(¯ h ) = ζ ( g ) ζ (¯ g ) χ ( h/sg ) χ (¯ h/s ¯ g ) · s (cid:48) ( ϕg ) s (cid:48) ( ϕ ¯ g ) , with g := πh and ¯ g := π ¯ h , which implies Φ : H → H (cid:48) is a homomor-phism iff χ (cid:0) h ¯ h/s ( g ¯ g ) (cid:1) · s (cid:48) ( ϕg ϕ ¯ g ) equals χ ( h ¯ h/sg s ¯ g ) δ · s (cid:48) ( ϕg ) s (cid:48) ( ϕ ¯ g ) ,where we set δ := d ( ζ )( g, ¯ g ) = ζ ( g ) ζ (¯ g ) /ζ ( g ¯ g ) . Multiplying throughby χ ( sg s ¯ g/h ¯ h ) δ − ∈ C (cid:48) , this is equivalent to χ ∗ ψ ( g, ¯ g ) δ − · s (cid:48) ( ϕg ϕ ¯ g ) being equal to s (cid:48) ( ϕg ) s (cid:48) ( ϕ ¯ g ) . By definition of the quotients for E (cid:48) , thisis in turn equivalent to ϕ ∗ ψ (cid:48) ( g, ¯ g ) = χ ∗ ψ ( g, ¯ g ) δ − or δ = χ ∗ ψϕ ∗ ψ (cid:48) ( g, ¯ g ) , aswas to be shown. (cid:3) It is clear that the setting of
SES sg comprises the common setting of central extensions of abelian groups → C ι → H π → G → , where C and G are abelian groups but where H may be any (possibly non-abelian) group such that C ≤ (cid:90) ( H ) . The setting SES sg generalizesthis in two respects: We allow H and G to be semigroups (while we haveseen that C is automatically a group), and we require only a compatiblefree action of C × H → H . The latter is induced by setting c · h := ι ( c ) h .Note that compatibility (as well as orbit commutativity) follows from C being central. Writing SES for the category of central extensions ofabelian groups as defined in § (cid:74) ?? (cid:75) , it is also clear that the morphismsof the latter become morphisms in SES sg . Thus one sees that SES is a (non-full) subcategory of
SES sg , and we shall identify the corre-sponding exact sequences along with their morphisms. In this sense,Proposition 68 is in fact a special case of Proposition (cid:74) ?? (cid:75) .The category SES sg arises naturally from function fields such as theones generated by Gaussians. The key to understanding the connectioninvolves some valuation theory [29, §9], which we briefly recall for fixingterminology. Given a field (cid:75) and an additively written totally orderedgroup (cid:71) , a valuation on (cid:75) with value group (cid:71) is a map ν : (cid:75) → (cid:71) ∪ {∞} such that ν ( (cid:75) × ) = (cid:71) and ν (0) = ∞ ,ν ( f f (cid:48) ) = ν ( f ) + ν ( f (cid:48) ) ,ν ( f + f (cid:48) ) ≥ min { ν ( f ) , ν ( f (cid:48) ) } N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 93
In that case, ( (cid:75) , ν ) is called a valuated field , (cid:65) ν := { f ∈ (cid:75) | ν ( f ) ≥ } its valuation ring , a ν := { f ∈ (cid:75) | ν ( f ) > } its maximal ideal ,and (cid:75) ν := (cid:65) ν / a ν its residue field .We use the notation (cid:71) ≥ g := { h ∈ (cid:71) | h ≥ g } , with similar definitionsfor (cid:71) >g , (cid:71) ≤ g and (cid:71)
The lower row of this diagram can be restricted to the positive degreesto yield(61) (cid:47) (cid:47) (cid:65) ∗ (cid:31) (cid:127) (cid:47) (cid:47) gr ν. ( a ) ¯ ν (cid:47) (cid:47) (cid:71) a (cid:47) (cid:47) , which is the exact sequence we instantiate for the Gaussians.In detail, we shall choose the valuation ring (cid:65) = (cid:71) ( R ) with maxi-mal ideal a = (cid:71) ( R ) , unit group (cid:65) ∗ = C × and residue field (cid:75) (cid:65) ∼ = C .The quotient field (cid:75) is a subfield of the field C ( (cid:69) ) introduced ear-lier; it consists of all rational functions in the g µ,ρ . The value group (cid:71) may be viewed as the additive subgroup R x + R x ≤ R [ x ] with lex-icographic order (first quadratic then linear term), with semigroup (cid:71) a = R > x + R x ∼ = (cid:71) • corresponding to a = (cid:71) ( R ) . It is easy tosee that in the present case gr ν ( (cid:65) ) ∼ = (cid:71) ( R ) with components (cid:65) ∼ = C and (cid:65) µ,ρ ∼ = [ g µ,ρ ] , so here ¯ ν essentially coincides with ν . Therefore weobtain also gr ν. ( (cid:65) ) ∼ = C × (cid:93) C × (cid:71) • . Finally, we have gr ν ( a ) ∼ = (cid:71) ( R ) and gr ν. ( a ) ∼ = C × (cid:71) • for the nonunital counterparts, which yields(62) (cid:47) (cid:47) C × (cid:31) (cid:127) (cid:47) (cid:47) C × (cid:71) • π (cid:47) (cid:47) (cid:71) • s (cid:108) (cid:108) (cid:47) (cid:47) for the exact sequence (61) encoding the quotient (cid:71) • ∼ = C × (cid:71) • / C × .Here π : C × (cid:71) • → (cid:71) • is the canonical projection cg µ,ρ (cid:55)→ ( µ, ρ ) while s : (cid:71) • → C × (cid:71) • the set-theoretic section of π given by ( µ, ρ ) (cid:55)→ g µ,ρ .Applying Lemma 67 to (62) and using the above standard section s : (cid:71) • (cid:44) → C × (cid:71) • , we obtain C × (cid:71) • ∼ = C × × γ • (cid:71) • , the pointwise Gauss-ian cocycle γ • := γ • ( µ , ρ ; µ , ρ ) = s ( µ , ρ ) s ( µ , ρ ) /s (cid:0) ( µ , ρ ) · ( µ , ρ ) (cid:1) in the form γ • = g µ ,ρ g µ ,ρ /g µ + ρ µ ,ρ + ρ so that(63) γ • = g ,ρ (cid:1) ρ ( µ − µ ) = exp (cid:16) − πρ ρ ( µ − µ ) ρ + ρ (cid:17) , which is the normalization constant for the ( µ, ρ ) parametrization with product law (52). Note that γ • = g µ ,ρ (cid:1) ρ ( µ ) = g µ ,ρ (cid:1) ρ ( µ ) We obtain from Lemma 67 also a characterization of the twistedsemigroup algebra C γ • [ (cid:71) • ] , namely as the complex vector space C ( (cid:71) • ) with multiplication induced by C × (cid:71) • . We obtain (cid:0) (cid:71) ( R ) , · (cid:1) ∼ = C γ • [ (cid:71) • ] ,so the Gaussians form a basis over C with(64) g µ ,ρ g µ ,ρ = γ • g µ,ρ with µ = µ + ρ µ , ρ = ρ + ρ and (63) for the multiplication law.It is well-known that the Gaussians are also closed under convolution (another reason for discarding the constant function g µ, = 1 ). Using At this point, the chosen parametrization may seem awkward, but it will comein handy when introducing the convolution structure and Fourier operator. ∗ normalized Gaussians, the result is again a normalized Gaussian whosemean and variance are given by, respectively, adding the original meansand variances. The explicit expression is(65) g µ ,ρ (cid:63) g µ ,ρ = γ (cid:63) g µ,ρ with µ = µ + µ , ρ = ρ (cid:1) ρ where γ (cid:63) := γ (cid:63) ( µ , ρ ; µ , ρ ) is the convolutive Gaussian cocycle to beintroduced below in (67).In analogy to the pointwise structure, we may also introduce thesemigroup (cid:71) (cid:63) = ( R , +) ⊕ ( R > , (cid:1) ) where ( R > , (cid:1) ) is the harmonic semi-group . In other words, the product law of (cid:71) (cid:63) is given by ( µ , ρ ) (cid:63) ( µ , ρ ) = ( µ + µ , ρ (cid:1) ρ ) , and we obtain(66) (cid:47) (cid:47) C × (cid:31) (cid:127) (cid:47) (cid:47) C × (cid:71) (cid:63) π (cid:47) (cid:47) (cid:71) (cid:63)s (cid:108) (cid:108) (cid:47) (cid:47) as the convolutive analog of (62). Here C × (cid:71) (cid:63) is the same as C × (cid:71) • as a set but endowed with the convolution (cid:63) inherited from (cid:83) ( R ) . Moreover,the maps π and s are also the same as in (62), but they are nowhomomorphisms with respect to the convolution rather than pointwiseproduct. In analogy to the pointwise structure, we obtain once morea twisted semigroup algebra (cid:0) (cid:71) ( R ) , (cid:63) (cid:1) ∼ = C γ (cid:63) [ (cid:71) (cid:63) ] with corresponding convolutive Gaussian cocycle (67) γ (cid:63) = √ ρ + ρ already used in (65) above.Altogether, (cid:0) (cid:71) ( R ) , (cid:63), · (cid:1) ∼ = C γ (cid:63) [ (cid:71) (cid:63) ] (cid:62) C γ • [ (cid:71) • ] is thus seen to be a(nonunital) twain subalgebra of the Schwartz class (cid:0) (cid:83) ( R ) , (cid:63), · (cid:1) . To geta Fourier singlet, we need closure under the Heisenberg action. As (cid:71) ( R ) is closed under translation, we have C [ (cid:71) ( R ) R ] = (cid:71) ( R ) ≤ C ( R ) , andProposition 50 yields the Gaussian closure (68) (cid:71) ( R ) = C ( R ) (cid:71) ( R ) ∼ = C ( R ) ⊗ C (cid:71) ( R ) , where C ( R ) is the C -linear span of the oscillating exponentials e α with frequency α ∈ R , as mentioned before Proposition 50. The ten-sor product structure of (68) shows that (cid:71) ( R ) has the complex basis (cid:0) e α g µ,ρ | α ∈ R , ( µ, ρ ) ∈ R × R > (cid:1) .Proposition 50 ensures on general grounds that (cid:71) ( R ) is a Heisenberg(plain) subalgebra of (cid:83) ( R ) under the product law (69) (cid:26) e α g µ ,ρ · e α g µ ,ρ = ¯ γ • e α g µ,ρ , ( ρ , µ , α ) · ( ρ , µ , α ) = ( ρ + ρ , µ + ρ µ , α + α ) =: ( ρ, µ, α ) N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 97 based on (64) and with the same cocycle ¯ γ • = γ • as in (63) becausethe product is direct.While it is not clear a priori that it is in fact a Heisenberg twainsubalgebra (cid:0) (cid:71) ( R ) , (cid:63), · (cid:1) ≤ ( (cid:83) ( R ) , (cid:63), · ) , this may be seen from the con-volution law (70) (cid:26) e α g µ ,ρ (cid:63) e α g µ ,ρ = ¯ γ (cid:63) e α g µ,ρ , ( ρ , µ , α ) (cid:63) ( ρ , µ , α ) = ( ρ (cid:1) ρ , µ + µ , α + ρ α ) =: ( ρ, µ, α ) based on (65), but with the normalization constant given by the ex-tended convolutive Gaussian cocycle ¯ γ (cid:63) := γ (cid:63) (cid:104) α − α | µ − ρ µ (cid:105) g , / ( ρ + ρ ) ( α − α ) (71) = √ ρ + ρ exp (cid:16) iτ ( α − α )( µ ρ − µ ρ ) / ( ρ + ρ ) − π ( α − α ) ρ + ρ (cid:17) as may be seen via the relation g µ,ρ e α = exp ( iτ αµ − πα /ρ ) g µ + iα/ρ,ρ ,where on the right-hand side we have brusquely usurped the Gaussiannotation with a complex-valued “mean”. For the record, let us alsostate the explicit Heisenberg action (72) x • e α g µ,ρ = exp ( − iτ αx ) e α g µ + x,ρ , ξ • e α g µ,ρ = e α + ξ g µ,ρ for x ∈ R and ξ ∈ R ∼ = R ∗ .We have already seen (Example 56) that g = g , is a fixed pointof the Fourier transform (cid:70) : L / ( R ) (cid:55)→ L / ( R ) or the correspondingrestriction (cid:70) : (cid:83) ( R ) (cid:55)→ (cid:83) ( R ) . Continuing from the singlet perspective(i.e. viewing (cid:70) as an endo morphism on the Schwartz class), the fixedpoint result (cid:70) g , = g , generalizes to(73) (cid:70) ( e α g µ,ρ ) = c e µ g − α, /ρ with c := (cid:104) α | µ (cid:105) / √ ρ = ρ − / e iταµ , as one confirms by applying the similarity theorem (mentioned afterProposition 44) to g ,ρ = S √ ρ g , and the Heisenberg relations (17) to g µ,ρ = µ • g ,ρ and e α g µ,ρ = α • g µ,ρ .Thus the Fourier transform on (cid:83) ( R ) restricts to an endomorphismon (cid:71) ( R ) , we obtain a twain subsinglet, which is clearly regular since (73)is invertible. Due to the prominent role played by the Gaussian normaldistribution, we call the resulting singlet the Gaussian singlet ¯ (cid:71) (cid:104) R | R (cid:105) . Proposition 69.
The Gaussian singlet ¯ (cid:71) (cid:104) R | R (cid:105) = [ (cid:71) ( R ) (cid:55)→ (cid:71) ( R )] isa regular twain subsinglet of (cid:83) (cid:104) R | R (cid:105) = [ (cid:83) ( R ) (cid:55)→ (cid:83) ( R )] . (cid:3) It may be interesting to view also the Gaussian closure (cid:71) ( R ) withits Fourier transform (cid:70) : (cid:71) ( R ) (cid:55)→ (cid:71) ( R ) from the viewpoint of exactsequences in SES sg . It is easy to see that the exact sequence (62) ∗ extends to(74) (cid:47) (cid:47) C × (cid:31) (cid:127) (cid:47) (cid:47) C × ¯ (cid:71) (cid:63) π (cid:47) (cid:47) (cid:70) (cid:15) (cid:15) ¯ (cid:71) (cid:63) (cid:47) (cid:47) (cid:102) (cid:15) (cid:15) (cid:47) (cid:47) C × (cid:31) (cid:127) (cid:47) (cid:47) C × ¯ (cid:71) • π (cid:47) (cid:47) ¯ (cid:71) • (cid:47) (cid:47) where ¯ (cid:71) • := (cid:71) • × C ( R ) is a direct product of monoids while C × ¯ (cid:71) • is semidirect (with the “same” cocycle as in (62) since the direct prod-uct does not contribute to cocycles). The elements of ¯ (cid:71) may be takenas triples ( ρ, µ, α ) ∈ (cid:71) • × R , on which the Fourier reflex operatesby (cid:102) ( ρ, µ, α ) = (1 /ρ, − α, µ ) . Thus we have (cid:102) = i × j with the iso-morphism i : ( R > , (cid:1) ) ∼ −→ ( R > , +) and the tilt map j : R → R discussed in §2.3. The latter makes sense because the parameters ( µ, α ) ∈ R = G ⊕ Γ may be seen as position/momentum pairs with G acting naturally on µ and Γ on α .By Proposition 68, the semigroup homomorphism (cid:102) determines (cid:70) up to the -chain ζ ∈ C ( ¯ (cid:71) (cid:63) , C × ) given by the constant c of (73)as ζ ( ρ, µ, α ) = (cid:104) α | µ (cid:105) / √ ρ . By a straightforward calculation using theconvolution law (70) as well as the (extended) cocycles (71) and (63),one may verify that d ζ ( ρ µ α ; ρ µ α ) = ζ ( ρ µ α ) ζ ( ρ µ α ) /ζ ( ρ µ α (cid:63) ρ µ α ) is given by ¯ γ (cid:63) / (cid:102) ∗ ¯ γ • = (cid:104) ρ (cid:48) α | µ (cid:105)(cid:104) ρ (cid:48) α | µ (cid:105) / (cid:104) α | ρ (cid:48) µ (cid:105)(cid:104) α | ρ (cid:48) µ (cid:105) as re-quired in Proposition 68.4.2. The Gelfond Field for Coefficients.
As we have seen in §4.1,the normal Fourier singlet ¯ (cid:71) (cid:104) R | R (cid:105) leads to rational expressions in e τξ ,where ξ is itself a rational expression in the parameters. In the nextsubsection we shall build up an algorithmic subdomain of ¯ (cid:71) (cid:104) R | R (cid:105) gen-erated by allowing only rational values for the parameters. We are thusled to consider Q (cid:0) e τξ | ξ ∈ Q ( i ) (cid:1) as coefficient field. In this small sub-section we will collate various number-theoretic facts about this field,which we define now in the equivalent form(75) Q π := Q (cid:0) e πξ | ξ ∈ Q ( i ) (cid:1) . In other words, Q π is generated by all powers of Gelfond’s constant e π having Gaussian rationals as exponents, and we shall thus refer to Q π as the Gelfond field . We analyze now the algebraic structure of thisfield, giving special emphasis to the important fact that it is well suitedfor the algorithmic treatment.
N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 99
The crucial observation is that Q π is built up from two rather dis-similar subfields, which we shall write as(76) Q tr := Q ( e πα | α ∈ Q ) and Q ab := Q ( e iπβ | β ∈ Q ) . Obviously, the Gelfond field is the compositum Q π = Q tr . Q ab . Itssecond factor Q ab is the maximal abelian extension [52, §8.1(j)] of therational field, obtained by adjoining all roots of unity to Q . In otherwords, we have the direct limit of cyclotomic fields(77) Q ab = (cid:91) n ∈ N > Q ( ζ n ) , where ζ n := e iτ/n is the n -th standard primitive root of unity. While Q ab is thus an algebraic extension field (in fact, a Galois extension), its com-panion Q tr is a transcendental extension of Q , for which it is easy toprovide a natural transcendece basis. Lemma 70.
The field Q tr is transcendental over Q with transcen-dence basis { e π } . It may be realized as the fraction field of the groupring Q [ η α | α ∈ Q ] (cid:44) → Q tr .Proof. For showing that { e π } is a transcendence basis, we must showthat e π is transcendental and that Q tr is algebraic over K := Q ( e π ) .The transcendence of Gelfond’s constant e π is well-known [13, §2.1], sowe need only prove the second claim. It suffices to check that everygenerator e πα ( α ∈ Q ) of Q tr is algebraic over K . Writing α = m/n with m ∈ Z and n ∈ N > , we note first that η := e π/n is algebraicover K since it is annihilated by x n − e π ∈ K [ x ] . But then e πα = η m isalgebraic over K as well since algebraic elements are closed under fieldoperations.Now let R := Q [ η α | α ∈ Q ] be the group ring having Q both as acoefficient field and as the underlying (additive) group. We show thatthe Q -linear map ι : R → Q tr , η α (cid:55)→ e πα is injective. Hence assume λ e πα + · · · + λ n e πα n = 0 for coefficients λ , . . . , λ n ∈ Q × and exponents α , · · · , α n ∈ Q . Wewrite the latter with common denominator N ∈ N > and with thenumerators k < · · · < k n so that we have λ η k + · · · + λ n η k n = 0 for η := e π/N . Multiplying this equation by η − k for k := min( k , . . . , k n ) ,we may ensure that k < · · · < k n . But then η is annihilatedby λ + λ x + · · · + λ n x n ∈ Q [ x ] , so that η and hence e π = η N isalgebraic, contradicting Gelfond’s result. We conclude that ι is indeedinjective, so R is an embedded Q -subalgebra of the field Q tr and thusin particular an integral domain. So we may form the fraction field ¯ R
00 MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ Q π Q tr0 Q ab0 Q ( e π ) Q Figure 4.
Subfields of the Gelfond Fieldof R , and we obtain the extended map ¯ ι : ¯ R → Q tr since (cid:54)∈ ι ( R × ) ;confer [50, Thm. III.4.5]. Being a homomorphism of fields, ¯ ι is clearlyinjective as well. But it is also surjective since we have e πα = ¯ ι ( η α ) foreach generator of the field Q tr = Q ( e πα | α ∈ Q ) . (cid:3) The group ring mentioned in the above lemma (as well as its imagein Q tr ) shall be written as Q [ Q , +] . The lemma also points the way togeneralizing this kind of extension: For arbitrary fields K, L of charac-teristic zero, one may introduce the group ring K [ L, +] with coefficientsin K over the additive group ( L, +) . Since the latter is always torsion-free and cancellative, one may infer [26, Prop. 4.20(b)] that the groupring K [ L, +] is an integral domain. Choosing in particular K = L ,we can introduce K tr as the fraction field of K [ K, +] , an intrinsic tran-scendental extension by a “saturated” set of exponential-like generators.The special case of K = Q is recovered via the identification mentionedin Lemma 70. In this case, Q tr is an ordered subfield of R and henceformally real.See Figure 4 for an extension diagram of the fields Q tr and Q ab . Theedges of this diagram are labelled by transcendence degrees (zero meansalgebraic extension): On the left branch we have tr . deg (cid:0) Q ( e π ) / Q (cid:1) = 1 by Gelfond’s result, tr . deg (cid:0) Q tr / Q ( e π ) (cid:1) = 0 by the proof of Lemma 70,and tr . deg( Q π / Q tr ) = 0 since Q π arises from Q tr by adjoing the alge-braic elements Q ab . For the right branch, we have tr . deg( Q ab / Q ) = 0 since the maximal abelian extension (77) is algebraic, and this forces tr . deg( Q π / Q ab ) = 1 because the transcendence degrees must add up N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 101 to . With this background knowledge, we can now establish the re-lationship between the subfields Q tr and Q ab of the Gelfond field Q π ,namely that there is essentially no interaction between them. Lemma 71.
The Q -algebras Q [ Q , +] and Q ab are linearly disjoint.Proof. According to [29, Prop. 11.6.1], it suffices to prove that thecanonical Q -basis { e πα | α ∈ Q } of Q [ Q , +] is also linearly independentover Q ab . Assuming otherwise, we may proceed as in the proof ofLemma 70 to obtain a relation of the form λ + λ η + · · · + λ n η n = 0 with η = e π/N and N ∈ N > , but with coefficients λ , . . . , λ n ∈ Q ab .This implies that η and hence e π = η N is algebraic over Q ab . Since Q π has { e π } for a transcendence basis, the whole Gelfond field Q π is thenalgebraic over Q ab , contradicting the extension relations establishedabove (Figure 4). (cid:3) From the results of the preceding lemmata, we can now provide afairly explicit description of the Gelfond field in terms of a tensor prod-uct.
Proposition 72.
Up to isomorphism, the Gelfond field Q π is given asthe fraction field of Q ab [ Q , +] .Proof. We have Q ab [ Q , +] ∼ = Q [ Q , +] ⊗ Q ab as Q -algebras, and thistensor product is in turn isomorphic [29, Prop. 5.4.2] to the Q -algebra A generated by Q [ Q , +] and Q ab . Hence it suffices to show that Q π is thefraction field ¯ A of A . It is clear that ¯ A ≤ Q π since Q [ Q , +] , Q ab ≤ Q π .For the converse, it suffices to check that each generator e πξ (cid:0) ξ ∈ Q ( i ) (cid:1) is contained in A , which is obvious since ξ = α + βi with α, β ∈ Q sothat e πξ = e πα e iπβ with e πα ∈ Q [ Q , +] and e iπβ ∈ Q ab . (cid:3) For obtaining a fully algorithmic description, we have to providecanonical forms. This is possible by using direct limits with respect toinclusions, similar to the maximal abelian extension (77). In the lattercase, basic theory [7, p. 187] suggests restricting to the cyclotomicfields Q ( ζ n ) with n (cid:54)≡ for avoiding duplication, and then Q ( ζ n ) ≤ Q ( ζ n (cid:48) ) holds true iff n | n (cid:48) . In the case of the transcendentalextension, we have(78) Q tr = (cid:91) n ∈ N > Q ( e π/n ) on the set-theoretic level, which may again be interpreted as a directlimit as follows.
02 MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ Lemma 73.
We have a direct limit (78) with respect to the inclusions Q ( e π/n ) ≤ Q ( e π/n (cid:48) ) , which hold true iff n | n (cid:48) . In fact, for any algebraicsubextension Q ≤ K ≤ Q π , one has K ( e π/n ) ≤ K ( e π/n (cid:48) ) iff n | n (cid:48) .Proof. The implication from right to left is clear. Hence let us assume e π/n ∈ K ( e π/n (cid:48) ) to show n | n (cid:48) . Then we have e π/n = r ( e π/n (cid:48) ) for somerational function r ( x ) = p ( x ) /q ( x ) ∈ K ( x ) with p ( x ) and q ( x ) coprime.Setting η := e π/nn (cid:48) , this is equivalent to q ( η n ) η n (cid:48) = p ( η n ) . Since η istranscendental over K , the evaluation homomorphism K ( x ) (cid:16) K ( η ) , x (cid:55)→ η is injective so that we have also q ( x n ) x n (cid:48) = p ( x n ) . Takingdegrees, we obtain n deg( q ) + n (cid:48) = n deg( p ) and hence n | n (cid:48) .For showing that the corresponding direct limit is indeed the sameas the union (78), it suffices to verify the following universal prop-erty: Given a family of homomorphisms (cid:0) f n : Q ( e π/n ) → K (cid:1) n ∈ N toan arbitrary field K with the coherence constraints f n (cid:48) | Q ( e π/n ) = f n whenever n | n (cid:48) , there is a unique homomorphism λ : Q tr → K suchthat f n = λ | Q ( e π/n ) . The latter condition determines λ on the subalge-bra Q [ Q , +] ≤ Q tr as λ ( η πα ) = f n ( η m/n ) , where we have set η := e π/n and where α = m/n ∈ Q is written in terms of m ∈ Z and n ∈ N > .This is well-defined since m/n = m (cid:48) /n (cid:48) implies f n ( η m/n ) = f nn (cid:48) ( η mn (cid:48) /nn (cid:48) ) = f nn (cid:48) ( η m (cid:48) n/nn (cid:48) ) = f n (cid:48) ( η m (cid:48) /n (cid:48) ) by the coherence constraints, hence also λ ( η m/n ) = λ ( η m (cid:48) /n (cid:48) ) as re-quired. But then λ is also determined on Q tr since this is the fractionfield of Q [ Q , +] by Lemma 70 so that λ ( p/q ) = λ ( p ) /λ ( q ) for any frac-tion p/q ∈ Q tr with p, q ∈ Q [ Q , +] and q (cid:54) = 0 . Thus we have establishedexistence and uniqueness of the homorphism λ : Q tr → K . (cid:3) We may now put together the two direct limits into a single one,which correspondingly has the form(79) Q π = (cid:91) m ∈ N > (cid:91) n ∈ N > Q ( e iπ/m , e π/n ) with the respect to the expected natural inclusion maps. Proposition 74.
We have the direct limit (79) with respect to theinclusions Q ( e iπ/m , e π/n ) ≤ Q ( e iπ/m (cid:48) , e π/n (cid:48) ) , which hold true iff m | m (cid:48) and n | n (cid:48) .Proof. The implication from right to left is again clear, so assumethe inclusion Q ( e iπ/m , e π/n ) ≤ Q ( e iπ/m (cid:48) , e π/n (cid:48) ) . Since linear disjoint-ness is preserved under subalgebras [29, §11.6], we see that the groupring Q [ e π/n ] ≤ Q [ Q , +] and the field Q ( e iπ/m ) ≤ Q ab are also linearlydisjoint. As in Proposition 72, it follows that Q ( e iπ/m (cid:48) , e π/n (cid:48) ) is the N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 103 fraction field of the Q -algebra A := Q ( e iπ/m (cid:48) )[ e π/n (cid:48) ] , which clearly has Q ( e iπ/m (cid:48) ) -basis { e ( k/n (cid:48) ) π | k ∈ Z } .Now consider e iπ/m ∈ Q ( e iπ/m (cid:48) , e π/n (cid:48) ) . Any e iπ/m (cid:54)∈ Q ( e iπ/m (cid:48) ) wouldbe a nonconstant element of the transcendental extension K ( e π/n (cid:48) ) ofthe field K := Q ( e iπ/m (cid:48) ) , which implies [112, p. 217] that e iπ/m istranscendental over K and hence over Q ; but e iπ/m is in fact alge-braic. Hence we see that e iπ/m ∈ Q ( e iπ/m (cid:48) ) , which forces m | m (cid:48) by theobservations before (78). We have also e π/n ∈ Q ( e iπ/m (cid:48) , e π/n (cid:48) ) , whichimplies K ( e π/n ) ≤ K ( e π/n (cid:48) ) for K := Q ( e iπ/m (cid:48) ) . By Lemma 73, thisyields n | n (cid:48) .The proof of the direct limit statement (79) proceeds as for Lemma 73.Hence let (cid:0) f m,n : Q ( e iπ/m , e π/n ) → K (cid:1) m,n ∈ N be any family of homo-morphisms to some field K , satisfying the corresponding coherenceconstraints f m (cid:48) ,n (cid:48) | Q ( e iπ/m ,e π/n ) = f m,n in case of m | m (cid:48) and n | n (cid:48) . Wemust show that there is a unique homomorphism λ : Q π → K suchthat f m,n = λ | Q ( e iπ/m ,e π/n ) . By its definition (75), the Gelfond field Q π is the rational function field in the transcendental generators e πξ with ξ = α + βi and α, β ∈ Q . Therefore any a ∈ Q π can be writtenas a = r ( e πξ , . . . , e πξ N ) for a rational function r ∈ Q ( x , . . . , x N ) and ξ j = k j n j + l j m j i ( j = 1 , . . . , N ) ; in this case we set λ ( a ) := r (cid:0) f m ,n ( e πξ ) , . . . , f m N ,n N ( e πξ N ) (cid:1) . Note that λ is well-defined thanks to the coherence constraints onthe f m,n . By its construction, it is also clear that λ satisfies the requiredrestriction property λ | Q ( e iπ/m ,e π/n ) = f m,n . Moreover, λ is uniquely de-termined by this condition since Q π is covered (set-theoretically) bythe component fields Q ( e iπ/m , e π/n ) . We have thus established the uni-versal property, and (79) is indeed a direct limit. (cid:3) The practical significance of Proposition 74 is the following. Anyterm T denoting an element in Q π can be located in some componentfield K m,n := Q ( e iπ/m , e π/n ) , and all possible choices of m are multiplesof each other; likewise for the choices of n . Choosing the minimal m and n , we can thus rewrite the given term in the form of a rationalfunction T (cid:48) in only e iπ/m and e π/n . The transformation T (cid:55)→ T (cid:48) isthen clearly a canonical simplifier [27], provided we have a canonicalsimplifier for the fields K m,n .Canonical simplifiers on K m,n are readily available: As we have seenin the proof of Proposition 74, each field K m,n is the fraction field ofthe Laurent polynomial ring K m [ η, η − ] in the indeterminate η := e π/n over the coefficient field K m = Q ( ζ ) , where ζ := e iπ/m is the m -th
04 MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ standard primitive root of unity. Arithmetic is K m is straightforwardsince it is an algebraic extension of Q with basis { , ζ, . . . , ζ d − } anddimension given by the Euler totient function as d := Φ( n ) . Canonicalforms in the Laurent polynomial ring K m [ η, η − ] can be achieved e.g.by expanding/reducing fractions (multiplying by a coefficient from K m and a suitable power of η ) such that numerator and denomoninator arepolynomials in η with minimial degree, and such that the denominatoris monic. Computing field operations with any representatives andsubsequent reduction to canonical form establishes [27, p. 13] that K m,n and thus also Q π is an effective field .As we shall see below (Lemma 76), the Gelfond field Q π is indeedsufficient for the basic operations of the normal Fourier singlet ¯ (cid:71) (cid:104) R | R (cid:105) .For the additional action of the Weyl algebra, however, we shall needthe extended Gelfond field Q π ( π ) in (88) since powers of π are croppingup when differentiating gaussians. Hence we need to ensure that wemay still treat π as a transcendental indeterminate when computingover Q π . Fortunately, this is the case. Proposition 75.
The number π is transcendental over Q π .Proof. Suppose π is algebraic over Q π . Since { e π } is a transcendencebasis of Q π by Proposition 74, this implies that the set { π, e π } is al-gebraically dependent. But this contradicts a well-known result byNesterenko [72, §1.5.7]. (cid:3) The Rational Fourier Singlet of Gaussians.
In order to iso-late a computable Fourier singlet from the uncountable Schwartz sin-glet (cid:83) (cid:104) R | R (cid:105) , we shall need two restrictions: • We restrict the Heisenberg action H R × (cid:83) ( R ) → (cid:83) ( R ) fromthe original Heisenberg group H R := TR (cid:111) R to the sub-group H Q := T Q Q (cid:111) Q , where T Q ∼ = Q / Z is the torsionsubgroup of T . Note that T Q consists of all roots of unityso that the extension field of Q generated by T Q is just themaximal abelian extension Q ab ≤ Q π considered in §4.2. Asit is clear that H Q is a Heisenberg group in the sense of Def-inition 2, we may refer to it as the (one-dimensional) rationalHeisenberg group . • The scaling parameter ρ of the Gaussians g µ,ρ is also restriced torational numbers. It is then clear that the action of the rationalHeisenberg group H Q will only produce Q π -linear combinationsof e α g µ,ρ with the parameters α, µ, ρ all rational. Referring to N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 105 the multiplication laws for (cid:63) and (cid:12) in §(4.1), one checks imme-diately that they form a twain subalgebra of the Gaussian clo-sure (cid:71) ( R ) , which we call the rational Gaussian closure (cid:71) ( Q ) .It is furthermore clear that [ (cid:70) : (cid:71) ( Q ) (cid:55)→ (cid:71) ( Q )] is a Fourier subs-inglet of [ (cid:70) : (cid:71) ( R ) (cid:55)→ (cid:71) ( R )] = ¯ (cid:71) (cid:104) R | R (cid:105) since the Fourier operator (cid:70) in (73) creates only rational parameters from the given rational pa-rameters. We shall therefore refer to [ (cid:71) ( Q ) (cid:55)→ (cid:71) ( Q )] as the rationalGaussian singlet ¯ (cid:71) (cid:104) Q | Q (cid:105) . It should be emphasized that ¯ (cid:71) (cid:104) Q | Q (cid:105) is a computable Fourier singlet since the coefficient field Q π is computableand all operations (especially convolution, pointwise product, Heisen-berg action, Fourier operator) are algorithmic.Recall that the unit Gaussian in Stigler normalization is denoted by g := g , ∈ (cid:83) ( R ) . We claim that the rational Gaussian singlet ¯ (cid:71) (cid:104) Q | Q (cid:105) may also be characterized as the Fourier singlet generated by g in theSchwartz singlet (cid:83) (cid:104) R | R (cid:105) , of course over the rational Heisenberg ac-tion β Q : H Q × (cid:83) ( R ) → (cid:83) ( R ) . Lemma 76.
The rational Gaussian closure ¯ (cid:71) (cid:104) Q | Q (cid:105) is the smallestFourier subsinglet in (cid:83) (cid:104) R | R (cid:105) ∈ Fou ( β Q ) over the field Q π that con-tains the unit Gaussian g .Proof. For the moment, let us write (cid:83) g ≤ (cid:83) (cid:104) R | R (cid:105) for the small-est Fourier singlet containing g , meaning the intersection of all suchsinglets. Since the Fourier operator (cid:70) of (cid:83) (cid:104) R | R (cid:105) fixes g , we have (cid:83) g = [( g ) (cid:83) ( R ) (cid:55)→ ( g ) (cid:83) ( R ) ] , where ( g ) (cid:83) ( R ) is the Heisenberg twain alge-bra generated by g in (cid:83) ( R ) over H Q . It is clear that ( g ) (cid:83) ( R ) ≤ (cid:71) ( Q ) ,so it remains to prove the converse inclusion. But this follows from e α g µ,ρ = α • µ • g ρ , where g ρ := g ,ρ may be obtained from g via con-volution and pointwise multiplication in the following two equivalentways: Writing ρ = n/m ∈ Q , one has(80) ( g n ) (cid:63)m = ( mn m − ) − / g n/m and ( g (cid:63)m ) n = m − n/ g n/m as one may easily check. In other words, the pointwise and convolutivepowers coincide within a numerical factor—a property that we shalluse below. (cid:3) The rational Gaussian closure ¯ (cid:71) (cid:104) Q | Q (cid:105) is in fact a regular twainsubsinglet of (cid:83) (cid:104) R | R (cid:105) ∈ Fou ( β Q ) , and it turns out that it can be de-scribed as a quotient of the free Heisenberg twain algebra T ( S ) intro-duced in Proposition(27). As usual, we write T ( u ) for the case of asingleton S = { u } . This holds for plain Gaussians g µ,ρ , but not for linear combinations: Taking f := g , − g , , say, leads to h := ( f ) (cid:63) − ( f (cid:63) ) (cid:54) = 0 with | h , (1 / | > . .
06 MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ Quotients of a Heisenberg twain algebra A must be taken with re-spect to twain Heisenberg ideals , meaning twain ideals (ideals simul-taeously with respect to convolution and pointwise product) that areclosed under the Heisenberg action. Given any subset S ⊆ A , the twainHeisenberg ideal generated by S is of course the smallest Heisenbergtwain ideal (= intersection of all Heisenberg twain ideals) containing S ;we denote it by (cid:104) S (cid:105) .Using this setup, we can define the corresponding relations as theHeisenberg twain ideal (cid:104) (cid:82) u (cid:105) ⊂ T ( u ) generated by (cid:82) u = (cid:82) u (Γ , (cid:63) ) ∪ (cid:82) u ( G, · ) ∪ (cid:82) u ( (cid:63), · ) using the three relation sets to be given shortly. Anticipating the de-sired isomorphism, we introduce the abbreviation u ρ := √ mn m − ( u n ) (cid:63)m for ρ = n/m ∈ Q . Then the relations (cid:82) u (Γ , (cid:63) ) are gleaned from themultiplication law of (cid:63) given in §(4.1), appearing here as(81) (cid:82) u (Γ , (cid:63) ) := { ( α • u ρ ) (cid:63) ( α (cid:48) • u ρ (cid:48) ) − c α,α (cid:48) ,ρ,ρ (cid:48) (cid:0) αρ (cid:48) + α (cid:48) ρρ + ρ (cid:48) • u ρ (cid:1) ρ (cid:48) (cid:1) } with c α,α (cid:48) ,ρ,ρ (cid:48) := e − π ( α − α (cid:48) ) / ( ρ + ρ (cid:48) ) / (cid:112) ρ + ρ (cid:48) , where the parameters α, α (cid:48) , ρ, ρ (cid:48) range over Γ = Q ≤ T G (cid:111)
Γ = H β Q .Similarly, the relations (cid:82) u ( G, · ) , extracted from the multiplication lawof · , manifest themselves as(82) (cid:82) u ( G, · ) := { ( µ • u ρ ) · ( µ (cid:48) • u ρ (cid:48) ) − c µ,µ (cid:48) ,ρ,ρ (cid:48) (cid:0) µρ + µ (cid:48) ρ (cid:48) ρ + ρ (cid:48) • u ρ + ρ (cid:48) (cid:1) } with c µ,µ (cid:48) ,ρ,ρ (cid:48) := e − π ( ρ (cid:1) ρ (cid:48) )( µ − µ (cid:48) ) . We notice the intriguing symmetry between (81) and (82), ultimatelydue to the convolution theorem (but note the distinct relative posi-tion of the primes in the two Heisenberg actors). The last relationset (cid:82) u ( (cid:63), · ) comes from the commutation (80) of convolution and point-wise powers, yielding(83) (cid:82) u ( (cid:63), · ) := { ( u (cid:63)m ) n − (cid:112) n m − /m n − ( u n ) (cid:63)m | n ∈ Z , m ∈ Z > } . Now we can state the promised isomorphism that characterizes ¯ (cid:71) (cid:104) Q | Q (cid:105) explicitly as a quotient of the free twain Heisenberg algebra. Theorem 77.
We have ¯ (cid:71) (cid:104) Q | Q (cid:105) ∼ = T ( u ) / (cid:104) (cid:82) u (cid:105) via the Heisenberg iso-morphism induced by g ↔ u .Proof. It is sufficient to show that { α • µ • u ρ | ( µ, α ) ∈ G × Γ , ρ ∈ Q } forms a system of mutually incongruent representatives whose classesare a Q π -linear basis of T ( u ) / (cid:104) (cid:82) u (cid:105) . For then e α g µ,ρ ↔ α • µ • u ρ Though G = Γ = Q as sets, we write α ∈ Γ and µ ∈ G to identify µ = 1( µ, and α = 1(0 , α ) , respectively, via the standard embeddings in H β Q . N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 107 will deliver the desired isomorphism. Indeed, it is clearly a Q π -linearisomorphism, it respects the Heisenberg action since e α g µ,ρ = ( µ, α ) • g ρ in ¯ (cid:71) (cid:104) Q | Q (cid:105) , and it is also a twain homomorphism: We have factoredout the corresponding relations (81) and (82), which yield the requiredmultiplication laws for (cid:63) and · when combined with the scalar/operatorproperties (see Definition 18) valid in T ( u ) .Oriented from left to right, we view the relations (cid:104) (cid:82) u (cid:105) as a termrewriting system over the signature [ AlgH ( β Q ) u ] introduced in thealternative proof of Proposition 27. Due to space limitations, we canonly sketch the rest of the proof. We consider (cid:63) and · modulo AC(associativity+commutativity), identifying unary products with theirarguments and nullary ones with ∈ Q π . Moreover, we identify thetorus elements c ∈ T Q < H β Q with the field scalars c ∈ Q π , so theHeisenberg actors are essentially given by ( µ, α ) ∈ Q × Q . Note thatthis rewrite system is ground (with u being a constant since it is notsubject to substitution). It is terminating as one can see by taking thelexicographic path order [11, §5.4.2] with u > · > (cid:63) > • on monomials(i.e. terms not containing field scalars or + ). As usual, this extendsto a Noetherian quasi-order on Q π -linear combinations of monomials;confer Theorem 5.12 in [15].The formal arguments in the relations (cid:82) u (Γ , (cid:63) ) , (cid:82) u ( G, · ) , (cid:82) u ( (cid:63), · ) of (81), (82), (83) indicate the joint occurrence of the correspondingsymbols in the redexes; this reveals that there is no overlap between therewrite rules. Hence the rewrite system is confluent, and we concludethat every element of T ( u ) has a unique normal form [11, Lem. 2.1.8].Since all elements α • µ • u ρ ∈ T ( u ) are irreducible, they must be mutuallyincongruent [11, Thm. 2.1.9]. Any Q π -linear combination of them isalso irreducible and hence clearly nonzero; this means that the α • µ • u ρ are Q π -linearly independent as elements of T ( u ) / (cid:104) (cid:82) u (cid:105) . It remains toshow that every term in T ( u ) reduces to a Q π -linear combination ofirreducible normal forms, which then implies that the α • µ • u ρ actuallyform a Q π -basis.For seeing that every normal form ϕ of T ( u ) is in the span of thenormal forms α • µ • u ρ , it suffices to expand ϕ into a Q π -linear com-bination of monomials and then reduce each of them to normal form.We may view the monomial as a tree of alternating [ (cid:63) ] and [ · ] nodesdecorated by Heisenberg actors (identifying the case of no actors withthe action of ∈ H β Q ). All leaves are occurrences of u . Let us speakof a [ (cid:63) ] tree if the root is a [ (cid:63) ] node (and hence every even tree levelconsists of [ (cid:63) ] nodes) and let us speak of a [ · ] tree otherwise. Startingfrom the leaves, the relations (cid:82) u (Γ , (cid:63) ) and (cid:82) u ( G, · ) together with the
08 MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ scalar/operator properties from Definition 18 serve to eliminate Heisen-berg actors from internal nodes: The u ρ + ρ (cid:48) , u ρ (cid:1) ρ (cid:48) in the correspondingright-hand monomials are trees without internal Heisenberg actors, sothese monomials have Heisenberg actors only in their roots. After ex-hausting these rewrite steps, we are left with α • µ • U , where U is eitherof the form u n/m = ( u n ) (cid:63)m or of the form ( u (cid:63)m ) n . In the former case,we are done; in the latter case, we apply (cid:82) u ( (cid:63), · ) for reducing to theformer case. (cid:3) The above proof is probably not as concise as it should be. Itwould be preferrable to use something like Gröbner(-Shirshov) bases,extended to the case of “twain polynomials”, for dealing with the twainpolynomial ideal (cid:82) u . This would be interesting to develop in futurework. It might also be worthwhile to consider enhanced rewrite ap-proaches that incorporate Gröbner bases such as [12].The algebraic description of (cid:71) ( R ) or (cid:71) ( Q ) may appear insignificantat first, but it should be pointed out that in some sense the Gaus-sians contain the whole essence of classical Fourier analysis: They areknown [110, Thm. 2.2] to be dense in L ( R ) . In the context of ourpresent approach, we view the Gaussian singlets (cid:71) ( R ) and (cid:71) ( Q ) as abase for bootstrapping algebraic hierarchies of more involved Fouriersinglets amenable to Symbolic Computation.4.4. Holonomic Fourier Extensions.
A distribution s ∈ (cid:68) ( R n ) iscalled holonomic if it is defined through a maximally overdeterminedset of polynomial PDEs. More precisely, writing I s = { T ∈ A n ( C ) | T s = 0 } for its annihilation ideal, one requires the quotient mod-ule A n ( C ) /I s to be holonomic [19, Def. 7.2.1], [121, §2.4]. The collectionof all holonomic distributions is known as the Bernstein class (cid:66) (cid:48) ( R n ) .It is clear [19, Prop. 2.2] that (cid:66) (cid:48) ( R n ) is then an A n ( C ) -submodule of (cid:68) ( R n ) .It is an important fact [19, Prop. 2.3] that a tempered distribution s ∈ (cid:83) ( R n ) belongs to (cid:66) (cid:48) ( R n ) iff (cid:70) s ∈ (cid:83) ( R n ) does. This means theFourier operator (cid:70) is a C -linear automorphism on the A n ( C ) -module (cid:72) (cid:48) ( R n ) := (cid:66) (cid:48) ( R n ) ∩ (cid:83) (cid:48) ( R n ) . The Dirac distributions δ ξ ( ξ ∈ R n ) are clearly contained in (cid:72) (cid:48) ( R n ) , and so are their Fourier transforms χ ξ ( x ) = e iτξ · x . Since (cid:68) (cid:48) ( R n ) is a module over C ∞ ( R n ) , the productof any holonomic distribution s ∈ (cid:72) (cid:48) ( R n ) with χ ξ ∈ C ∞ ( R n ) is well-defined in (cid:68) (cid:48) ( R n ) , and since both s and χ ξ are holonomic, so is theirproduct [121, Prop. 3.2]. This implies that (cid:72) (cid:48) ( R n ) is closed under Writing D := A n ( C ) , this is usually called a “ D -module”. We refrain here fromthis terminology so as to avoid confusion with the space of bump functions (cid:68) ( R n ) . N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 109 pointwise multiplication by χ ξ , which is identical to the action of theHeisenberg operator ξ ∈ H ( (cid:36) ) . By applying the Fourier operator (cid:70) : (cid:72) (cid:48) ( R n ) → (cid:72) (cid:48) ( R n ) , we obtain also closure under Heisenberg scalarswhile the torus action T ≤ C × is anyway trivial. Thus (cid:72) (cid:48) ( R n ) is aHeisenberg submodule of (cid:83) (cid:48) ( R n ) , and it inherits the compatible Weylaction. We summarize these facts on the space (cid:72) (cid:48) ( R n ) of holonmicdistributions as follows. Proposition 78.
The holonomic distributions form a regular slain sub-singlet (cid:72) (cid:48) (cid:104) R n | R n (cid:105) of (cid:83) (cid:48) (cid:104) R n | R n (cid:105) with compatible Weyl action. Writing (cid:72) ( R n ) := (cid:66) (cid:48) ( R n ) ∩ (cid:83) ( R n ) for the holonomic Schwartz class ,the Fourier operator (cid:70) : (cid:72) (cid:48) ( R n ) → (cid:72) (cid:48) ( R n ) clearly restricts furtherto a C -linear automorphism (cid:72) ( R n ) → (cid:72) ( R n ) . As the intersectionof Heisenberg modules (cid:72) (cid:48) ( R n ) and (cid:83) ( R n ) , the holonomic Schwartzclass (cid:72) ( R n ) is clearly a Heisenberg module itself. But it is moroevera Heisenberg twain algebra since pointwise and convolution productspreserve holonomicity [121, Prop. 3.2, 3.2 ∗ ]. Altogether we obtain thefollowing facts. Proposition 79.
The holonomic Schwartz class (cid:72) (cid:104) R n | R n (cid:105) forms aregular twain subsinglet of (cid:83) (cid:104) R n | R n (cid:105) with compatible Weyl action. Holonomic distributions and Schwartz functions are very suitablefor
Symbolic Computation since one can use normal forms for decidingequalities. To be precise, one does not have a canonical simplifier buta a normal simplifier in the sense of [27, §1], so deciding equalities isreduced to zero recognition; see [121, §4.1] and Algorithm Z of [36, B.2].The crucial tool for computing the Fourier operator (cid:70) symbolically isthe compatible Weyl action: Applying the differentiation laws (43) al-lows us to extract the defining PDEs of (cid:70) s from those of a holonomicdistribution or Schwartz function, as also pointed out in the paragraphpreceding [121, Prop. 3.2 ∗ ]. Sums, pointwise and convolution productsare all effectively computable [80, Thm. 6.6]. When dealing with holo-morphic functions (which form a valuation ring), one may also takeadvantage of the valuation techniques from §4.1. Example 80.
As a warmup, let us rederive the Fourier transform (73)of the modulated Gaussians e α g µ,ρ ∈ (cid:71) ( R ) from the holonomic perspec-tive. By differentiating the function once, we immediately obtain itsannihilator T α,µ,ρ = ∂ + τ ρ x − τ ( ρµ + iα ) ∈ A ( C ) . Note that T := T α,µ,ρ is uniquely determined (if chosen monic, for any ordering with ∂ > x ),and that α, µ, ρ can then be read off via(84) ρ = τ [ x ] T, µ = − τρ (cid:60) [1] T, α = − τ (cid:61) [1] T,
10 MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ where [ ∂ j x k ] denotes the coefficient of ∂ j x k . Applying the Fourier trans-form f of (42), we see that (cid:70) ( e α g µ,ρ ) is annihilated by iρ − (cid:91) T α,µ,ρ = ∂ + τρ x + τρ ( α − iρµ ) , which via (84) yields immediately (cid:70) ( e α g µ,ρ ) = e ˆ α g ˆ µ, ˆ ρ with ˆ ρ = 1 /ρ , ˆ α = µ , ˆ µ = − α as in (73). Thus we see that the holonomic approachrecovers the Fourier transform up to the overall constant denoted by c in (73).The constant c = c α,µ,ρ may be determined as follows. By our choiceof normalization for the Gaussians g µ,ρ , the L norm square of e α g µ,ρ is given by / √ ρ . Since (cid:70) preserves the L norm square, we ob-tain | c α,µ,ρ | = (cid:112) ˆ ρ/ρ = 1 /ρ , and it remains only to determine thephase dependence arg c α,µ,ρ . Note first that (cid:70) g ,ρ = c , ,ρ g , /ρ im-plies arg c , ,ρ = 1 since the Fourier transform of a real-valued evensignal is again real valued and even [24, §2]. Together with | c | = 1 / √ ρ this fixes the value c , ,ρ = 1 / √ ρ . Applying the twist axiom (H )yields e α g µ,ρ = (0 , α ) • ( µ, • g ,ρ = (cid:104) α | µ (cid:105) ( µ, α ) • g ,ρ whose Fouriertransform is c α,µ,ρ e µ g − α, /ρ = (cid:104) α | µ (cid:105) ( µ, − α ) • (cid:70) g ,ρ by the equivariancelaws (16). Using again (cid:70) g ,ρ = c , ,ρ g , /ρ and the Heisenberg actionon spectra (23), we get c α,µ,ρ e µ g − α, /ρ = (cid:104) α | µ (cid:105) c , ,ρ e µ g − α, /ρ and thus c α,µ,ρ = (cid:104) α | µ (cid:105) c , ,ρ = (cid:104) α | µ (cid:105) / √ ρ as in (73). // Example 81.
Consider the quartic Gaussian g ( x ) := e − x ∈ (cid:72) ( R ) ,which is relevant in various applications ranging from computer graph-ics, neutral networks and data interpolation [23] to energy correla-tions in random matrices [40]. Obviously, g is annihilated by theWeyl operator T = ∂ + 4 x ∈ A ( C ) whose Fourier transform is givenby ( iτ ) ˆ T = 4 ∂ − τ x . Therefore the Fourier transform ˆ g ( x ) of g ( x ) satisfies the third-order equation g (cid:48)(cid:48)(cid:48) ( x ) − τ x ˆ g ( x ) = 0 . For fixingthe solution, we need three initial values ˆ g ( j ) (0) for j = 0 , , . Viathe differentiation laws (43), we have ˆ g ( j ) (0) = ( iτ ) j µ j with the mo-ments µ j = (cid:114) ∞−∞ g ( x ) x j dx . The general solution of the third-orderequation is hypergeometric and can be obtained by standard symbolicsoftware packages. Using Mathematica ® , one finds with integrationconstants named c , c , c ∈ C that(85) ˆ g ( x ) = c Φ / , / ( x ) + c ζx Φ / , / ( x ) + c ζ x Φ / , / ( x ) where ζ := τ ζ / with ζ = e iτ/ the primitive eighth root of unity, andwhere Φ a,b ( z ) := F (cid:2) a b | ( τ z/ (cid:3) denotes a (generalized) hypergeo-metric function with lower parameters a, b ∈ C \ Z − . Note that Φ a,b isan entire function of the complex variable z . N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 111
The moments are ( µ , µ , µ ) = (cid:0) / , , Γ(3 / (cid:1) , the deriva-tives of (85) by (cid:0) ˆ g , ˆ g (cid:48) (0) , ˆ g (cid:48)(cid:48) (0) (cid:1) = ( c , ζc , ζ c ) , hence the con-stants are given by ( c , c , c ) = (cid:0) / , , i Γ(3 / (cid:1) . Substitungthese in (85) and using Γ(3 /
4) Γ(5 /
4) = π √ , this yields the represen-tation ˆ g ( x ) = π Γ(3 / √ Φ / , / ( x ) − Γ(3 / π x Φ / , / ( x ) . ApplyingLegendre’s duplication Γ(3 / /
4) = π √ leads to(86) ˆ g ( x ) = γ Φ / , / ( x ) − π √ γ Φ / , / ( x ) x , where have set γ := Γ(1 / . Adapting to different normalizations, (86)agrees with [23, (7)] as ˆ g ( k/τ ) and with [40, (6)] in the form ˆ g ( x √ /τ ) .It is known that γ is transcendental. But more is true: Sinceby Nesterenko’s results one knows [72, §1.5.7] that { π, e π , γ } is alge-braically independent over Q , we can conclude that γ is in fact tran-scendental over Q π ( π ) . In §4.2 we have seen how one may computein the Gelfond field Q π . From (86) we see that it is sufficient to passto the bivariate rational function field Q π ( π, γ ) for working with theFourier transform ˆ g ( x ) . // In §4.3 have seen that the unit Gaussians g = g , plays a central rolein describing the rational Gaussian closure (cid:71) ( Q ) . It has the convenientproperty of being a fixed point (i.e. an eigenfunction with eigenvalue )of the Fourier operator (cid:70) : (cid:83) ( R ) → (cid:83) ( R ) . This is generalized by the Hermite polynomials , which are eigenvectors for the four possible eigen-values ± , ± i of (cid:70) .For doing so, we extend the Gaussian closure (cid:71) ( R ) ⊂ (cid:83) ( R ) to aHeisenberg twain subalgebra of (cid:83) ( R ) just large enough for accommo-dating the action of the Weyl algebra. Clearly, such an algebra is givenby(87) (cid:71) ( R ) := C [ x ] ⊗ C (cid:71) ( R ) , where the tensor product creates a nonunital C -algebra from the uni-tal C [ x ] and the nonunital (cid:71) ( R ) . We refer to (cid:71) ( R ) as the Gaussian D -module .Of course, we may also define its computable core, the rationalGaussian D -module (cid:71) ( Q ) , by making the obvious replacements in (87),thus defining(88) (cid:71) ( Q ) := ˜ Q [ x ] ⊗ ˜ Q (cid:71) ( Q ) , where ˜ Q := Q π ( π ) is the extended Gelfond field introduced beforeProposition 75. In the sequel, we shall stick to (cid:71) ( R ) for ease of presen-tation. Its (pointwise) algebra structure is clear from the defintion (87),
12 MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ but for its convolution product and Fourier operator are best under-stood via Hermite functions [105, §7.1], so let us now study these insome more detail. Example 82.
As in [105, Exc. 7.1.3], we use the so-called physicists’Hermite polynomials H n ( x ) := (2 x − ddx ) n (1) = ( − n e x d n dx n e − x andthe unit Gaussian g ( x ) = e − πx to define the scaled Hermite functions η n ( x ) = c n g ( x ) H n ( τ / x ) with normalization constant /c n = 2 n n ! √ π chosen to suit our conventions in this paper. Compared to the standardHermite functions ϕ n of [110, §7.2.1], which are orthonormal in L ( R ) ,we have η n ( x ) = ϕ n ( τ / x ) .Since (cid:70) = 1 , it is clear that (cid:70) has eigenvalues ± , ± i and that (cid:70) isan algebraic operator (an algebraic element of the linear operator al-gebra according to [86, Thm. 1.4.1]. The corresponding eigenfunctionsare the η n whose scaling was specifically chosen to ensure (cid:70) η n = i n η n .This is a standard result of Fourier theory. For example, the proofin [108, §7.6], with slightly different normalization, is essentially alge-braic: It involves only the ladder operators associated to the harmonicoscillator H = D + x , whose eigenvalues determine the annihila-tors T n = D + x − n +1 τ ∈ A ( C ) of the corresponding η n . Exceptfor the different sign convention for (cid:70) , the Fourier pairs [ η n (cid:55)→ i n η n ] coincide with [24, Exc. 8.25].As an algebraic operator with four simple eigenvalues ± , ± i , thespace (cid:71) ( R ) splits [86, Cor. 1.4.1] into the four associated eigenspaces E k = [ η n | n ≡ k (mod 4)] with the corresponding projectors P k .Over the larger space, the finite linear combinations inherent in thespans and projectors are relaxed to series subject to suitable growthconstraints [108, §7.6]. When the eigenvalues are n -th roots of unity(in [86, Exc. 1.4.1] the corresponding operators are named “involutionsof order n ”), the projectors associated to A may be expressed as aver-ages over the distinct iterates A k weighted by the eigenvalues λ k . Thesimplest example is the symmetrizer P = (cid:80) = (1 + A ) / along withthe corresponding antisymmetrizer P − = (1 − A ) / in the case n = 2 .In the present setting with n = 4 , we have a fourfold sum [86, (1.4.27)];from a purely analytic viewpoint such a decomposition is of course “notconsidered very interesting” [108, p. 132].What is more important, at least from an algebraic perspective, isthe fact that there are explicit formulae, such as (13.9) and Exc. 13.1.6in [9], for changing between the monomial to the Hermite basis in C [ x ] ,and hence beween the corresponding bases of (cid:71) ( R ) . // N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 113
For verifying that (cid:71) ( R ) is closed under (cid:70) and also computable (onthe Gaussian D -submodule, that is), we compute the action of (cid:70) on thestandard basis x k e α g µ,ρ . Let us first restrict to the case α = 0 . Using(42) and (73), we have (cid:70) ( x k g µ,ρ ) = ( iτ ) − k d k dx k ρ − / e µ ( x ) g , /ρ ( x ) . Atthis point, we need the repeated derivatives of Gaussians, which maybe expressed in terms of Hermite polynomials as(89) g ( j ) µ,ρ ( x ) = ( − j ( πρ ) j/ H j (cid:0) ( πρ ) / ( x − µ ) (cid:1) g µ,ρ ( x ) , as one can immediately check by induction on k . Using the Leibnizrule for repeated differentiation, (43), and e ( j ) µ = ( iτ µ ) j e µ yields(90) (cid:70) ( x k g µ,ρ ) = µ k √ ρ e µ ( x ) g , /ρ ( x ) k (cid:88) j =0 (cid:0) kj (cid:1)(cid:0) − πρµ (cid:1) j/ H j ( (cid:112) π/ρ x ) . This can be simplified by using the so-called Appell identity [92, §4.2.1],which in our settings is given by H k (cid:0) y + s (2 ν ) / (cid:1) = (cid:0) ν (cid:1) − k/ k (cid:88) j =0 (cid:0) kj (cid:1)(cid:0) ν (cid:1) j/ H j (cid:0) y (2 ν ) / (cid:1) s k − j . Substituting ν = − / µ πρ , y = ix /ρµ , s = 1 in this, (90) becomes(91) (cid:70) ( x k g µ,ρ ) = √ ρ (cid:0) i √ πρ (cid:1) k H k (cid:0)(cid:112) π/ρ ( x − iµρ ) (cid:1) e µ ( x ) g , /ρ ( x ) . Before passing on to the general case, it is expedient to introduce theHermite polynomials with variance ν as in [92, §4.2.1] by(92) H [ ν ] k ( x ) := (cid:0) ν (cid:1) k/ H k ( x/ √ ν ) . Using these, (91) is given by(93) (cid:70) ( x k g µ,ρ ) = ( i/ρ ) k √ ρ H [ ρ/τ ] k ( x − iµρ ) e µ ( x ) g , /ρ ( x ) , and now the unmodulated special case immediately generalizes via (16)to(94) (cid:70) ( x k e α g µ,ρ ) = (cid:104) α | µ (cid:105)√ ρ ( i/ρ ) k H [ ρ/τ ] k ( x + α − iµρ ) e µ ( x ) g − α, /ρ ( x ) , which at the same time generalizes (73).Closure of (cid:71) ( R ) under (cid:63) follows from s (cid:63) s = (cid:70) − ( (cid:70) s · (cid:70) s ) forall s , s ∈ (cid:71) ( R ) , where we may use (94) for computing (cid:70) as well as (cid:70) − = (cid:70) ◦ (cid:80) in the x k e α g µ,ρ basis. However, this may not be themost promising route for obtaining an explicit form of the convolutionlaw since products of translated Hermite polynomials with differentvariances tend to be cumbersome.
14 MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ Substituting directly into the definition of convolution, one gets theproduct p ( x ) := x k e α g µ ,ρ (cid:63) x k e α g µ ,ρ as(95) p ( x ) = (cid:104) α | x (cid:105) (cid:90) ∞−∞ (cid:104) α − α | y (cid:105) ( x − y ) k y k g x − µ ,ρ ( y ) g µ ,ρ ( y ) dy, using g µ ,ρ ( x − y ) = g x − µ ,ρ ( y ) and the definition of oscillating ex-ponentials. We observe that (95) has the form of a Fourier transformwith respect to y , and we may apply (64) for multipliying the two Gaus-sians in the integrand to obtain a Gaussian with paramters ˆ ρ := ρ + ρ and ˆ µ x := ρ x − ρ µ + ρ µ ρ + ρ . Thus we can simplify (95) to(96) p ( x ) = e α ( x ) g µ + µ ,ρ (cid:1) ρ ( x ) (cid:70) y (cid:0) ( x − y ) k y k g ˆ µ x , ˆ ρ ( y ) (cid:1) (∆ α ) , with ∆ α := α − α , and we can apply (93) to compute the Fouriertransform in (96) as (cid:70) y (cid:0) . . . (cid:1) (∆ α ) = k (cid:88) j =0 (cid:0) k j (cid:1) ( − k + j x j (cid:70) y (cid:0) y k − j g ˆ µ x , ˆ ρ ( y ) (cid:1) (∆ α )= ( i/ρ ) k g , / ˆ ρ (∆ α )ˆ ρ / (cid:104) ∆ α | ˆ µ x (cid:105) k (cid:88) j =0 (cid:0) k j (cid:1) ( − k ( i ˆ ρx ) j H [ˆ ρ/τ ] k − j (∆ α − i ˆ µ x ˆ ρ ) , where k := k + k . We have (cid:104) ∆ α | ˆ µ x (cid:105) e α ( x ) = (cid:104) α − α | µ − ρ µ (cid:105) e α ( x ) with the same compound frequency α := α + ρ α as for the convolutionproduct (70) without monomials. We thus obtain p ( x ) = ( − k ( i/ρ ) k ¯ γ (cid:63) Φ [ ρ/τ ] k ,k ( i ˆ ρx, ∆ α − i ˆ µ x ˆ ρ ) e α g µ,ρ , where ¯ γ (cid:63) is the extended convolutive Gaussian cocycle (71) and where(97) Φ [ ν ] m,n ( ξ, η ) := m (cid:88) j =0 (cid:0) mj (cid:1) ξ j H [ ν ] m + n − j ( η ) is a certain polynomial in ( ξ, η ) of bidegree ( m, n ) .For identifying this polynomial, we apply the methods of the umbralcalculus [92, §2.2], viewing differential operators as power series in theformal variable t , for which we may substitute ∂ ξ or ∂ η at will. Since theHermite polynomials H [ ν ] form an Appell sequence [92, §4.2] with gen-erator g ( t ) = e νt / , we have the relation H [ ν ] m + n − j ( η ) = e − ν∂ η / η m + n − j so that (97) immediately yields the nice representation(98) Φ νm,n ( ξ ) = e − ν∂ η / ( ξ + η ) m η n . Computationally speaking, the task of determining Φ [ ν ] m,n is in a sensealready achieved at this point: Expanding the exponential series in ∂ η N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 115 up to order m + n , one can determine every polynomial Φ [ ν ] m,n ( ξ, η ) ex-plicitly for any specific values of m, n and ν . We will provide a concretepractical calculation scheme below. But for theoretical purposes, how-ever, it is important to identify the precise “umbral species” of Φ [ ν ] m,n . Proposition 83.
The polynomials Φ [ ν ] m,n ( ξ, η ) ∈ Q ( η )[ ξ ] form an Appellsequence with Sheffer operator g ( t ) − = e ηt − νt / H [ ν ] n ( η − νt ) .Proof. Differentiating (98) with respect to ξ and with η = η , n = n fixed leads to ∂ ξ Φ [ ν ] m,n ( η ) = m Φ [ ν ] m − ,n ( η ) . Regarded as sequencein the polynomial ring Q ( η )[ ξ ] , this exhibits Φ m,n ( η ) as a Sheffersequence for (cid:0) t, g ( t ) (cid:1) according to [92, Thm. 2.3.7]. Since Appell se-quences are by definition [92, §2.3] Sheffer sequences for f ( t ) = t , thisestablishes the first claim. (We shall henceforth drop the subscript indicating the fixed values.)For identifying the Appell generator g ( t ) or, equivalently [92, (3.5.1)],the induced Sheffer operator g ( t ) − , we use the fact [92, §3.5] that thelatter maps the standard monomials ξ m ∈ Q ( η )[ ξ ] to the Appell poly-nomials Φ [ ν ] m,n ( ξ, η ) ∈ Q ( η )[ ξ ] . From (97) we have Φ [ ν ] m,n ( ξ, η ) = ∞ (cid:88) j =0 1 j ! H [ ν ] j + n ( η ) m j ξ m − j , which, together with m j ξ m − j = t j ξ m for t = ∂ ξ , yields the Shefferoperator in the form g ( t ) − = (cid:80) j H [ ν ] j + n ( η ) t j j ! ∈ Q ( η )[[ t ]] . Applying anindex shift and the relation ∂ nt t j j ! = t j − n ( j − n )! , this means g ( t ) − = ∂ nt ∞ (cid:88) j =0 H [ ν ] j ( η ) t j j ! = ∂ nt e ηt − νt / , where we have used the Hermite generating function [92, §4.2] for elim-inating the sum. For finishing the proof, one applies induction to show ∂ nt e ηt − νt / = e ηt − νt / H [ ν ] n ( η − νt ) . The base case follows from H [ η ]0 = 1 ,the induction step from the recurrence relation [92, (4.2.2)]. (cid:3) We have determined the Appell generator g ( t ) in so far as we knowits reciprocal g ( t ) − from Lemma 83 and can then apply the usualrecursive formula for finding the coefficients of g ( t ) . For actually com-puting g ( t ) , we would prefer an explicit representation. To this end,we shall employ the so-called generalized Feldheim identity [9, (13.47)]for computing powers of Hermite polynomials, which we quote here for
16 MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ easier reference with the following notational conventions: For a multi-index λ ∈ N m − we set | λ | i := λ + · · · + λ i − and | λ | := | λ | m . Notethat this implies | λ | = 0 . Lemma 84.
For k > , the k -th Hermite power is given by H [ ν ] n ( ζ ) k = (cid:88) λ ∈ Λ( n,k ) a λ H [ ν ] nk − | λ | ( ζ ) with coefficients a λ = k − (cid:89) i =1 (cid:18) nλ i (cid:19)(cid:18) ni − | λ | i λ i (cid:19) ν λ i λ i ! and Λ( n, k ) = { λ ∈ N k − | ∀ i =1 ,...,k − ≤ λ i ≤ min( n, ni − | λ | i ) } assummation range. We can now present the
Appell generator g ( t ) in a fairly explicit rep-resentation. Note that it does contain Hermite reciprocals, but unlikethe raw expression of g ( t ) , they do not involve the formal parameter t . Proposition 85.
The Appell generator g ( t ) of Φ [ ν ] m,n ( ξ, η ) ∈ Q ( η )[ ξ ] ,as treated in Lemma 83, is given by g ( t ) = ∞ (cid:88) l =0 l (cid:88) m =0 m (cid:88) k =0 (cid:88) λ ∈ Λ( n,k ) (cid:0) lm (cid:1)(cid:0) m +1 k +1 (cid:1) ( − k + l ν m ( nk − | λ | ) m a λ × H [ ν ] nk − m − | λ | ( η ) H [ − ν ] l − m ( η ) H [ ν ] n ( η ) k +1 t l l ! where Λ( n, k ) and a λ are as in Lemma 84.Proof. From Lemma 83, we have g ( t ) = e − ηt + νt / H [ ν ] n ( η − νt ) − , sothe first task it so find the reciprocal of H [ ν ] n ( η − νt ) . Since we aredealing with formal power series, it is sufficient to determine its Taylorcoefficients. To this end, we utilize the nice generic formula [64] foriterated derivatives of the reciprocal of a function that we may take tobe a power series h ∈ K [[ t ]] . Writing ∂ = ∂ t , this formula reads(99) ∂ m (1 /h ) = m (cid:88) k =0 (cid:0) m +1 k +1 (cid:1) ( − k ∂ m ( h k ) /h k +1 , and we apply it to h = H [ ν ] n ( η − νt ) . We substitute ζ = η − νt inLemma 84, whence we obtain ∂ m h k = (cid:80) λ a λ ( − ν ) m ( nk − | λ | ) m ˜ h with ˜ h := H [ ν ] nk − m − | λ | ( η − νt ) , by the usual differentation rule for Appell N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 117 sequences [92, Thm. 2.5.6] since H [ ν ] n is indeed Appell [92, §4.2.1]. Thuswe have for h ( t ) − the Taylor series ∞ (cid:88) m =0 m (cid:88) k =0 (cid:88) λ ∈ Λ( n,k ) (cid:0) m +1 k +1 (cid:1) ( − k + m ν m ( nk − | λ | ) m a λ H [ ν ] nk − m − | λ | ( η ) H [ ν ] n ( η ) k +1 t m m ! . Now we need only convolve this with the Taylor series of e − ηt + νt / = ∞ (cid:88) m =0 H [ − ν ] m ( − η ) t m m ! embodying the generating function of Hermite polynomials [92, §4.2.1],to end up with the formula given in the proposition (after a small signsimplification). (cid:3) As mentioned above, an explicit expression for the Appell genera-tor g ( t ) is certainly not necessary for “computing” the bivariate polyno-mials Φ [ ν ] m,n ( ξ, η ) . One straightforward method is the following recursivescheme . We note that Φ [ ν ] m, ( ξ, η ) = H [ ν ] m ( ξ + η ) and Φ [ ν ]0 ,n ( ξ, η ) = H [ ν ] n ( η ) ,latter relation following from (97) immediately, the former by the Ap-pell identity [92, Thm. 2.5.8]. Furthermore, we have the recurrence Φ [ ν ] m +1 ,n ( ξ, η ) = ξ Φ [ ν ] m,n ( ξ, η ) − Φ [ ν ] m,n +1 ( ξ, η ) , which can be obtained asa simple consequence of (98). Thus one may compute all polynomi-als Φ [ ν ] m,n with ( m, n ) ∈ { , . . . , M } for some fixed M > starting withthe boundary values for (0 , n ) and ( m, at the axes, then proceedingin diagonals with the stencil ( m, n ) ← { ( m − , n ) , ( m − , n + 1) } . Input : M ∈ Z > , ν ∈ R Output: (cid:0) Φ [ ν ] m,n (cid:1) m,n =0 ,...,M for m ← to M do Φ [ ν ] m, ← H [ ν ] m ( ξ + η ) endfor n ← to M do Φ [ ν ]0 ,n ← H [ ν ] n ( η ) endfor d ← to M dofor j ← to d − do Φ [ ν ] j,d − j ← ξ Φ [ ν ] j − ,d − j + Φ [ ν ] j − ,d − j +1 endend Algorithm 1: Calculation of Φ [ ν ] m,n
18 MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ The Hyperbolic Fourier Doublet.
Let us briefly look at a lastexample exhibiting explicit Fourier transforms. It is well-known thatthe hyperbolic secant is another fixed point of (cid:70) : (cid:83) ( R ) (cid:55)→ (cid:83) ( R ) . Inour normalization, (cid:70) s = s for the Schwartz function s ( t ) := sech( πt ) ;note that (cid:114) ∞−∞ s ( t ) dt = 1 . We are interested in the (plain) Fourierdoublet [ S (cid:55)→ Σ] generated by s in [ (cid:70) : (cid:83) ( R ) (cid:55)→ (cid:83) ( R )] qua Fourierdoublet. We shall here restrict ourselves to the trivial Heisenberggroup H ( β ) = 0 , which means in effect that we admit only s with-out adjoining its translates (see the remark below).In the case of hyperbolic—as opposed to trigonometric—functions,it is possible to restrict ourselves to real-valued functions. Indeed, thefunction s and all its powers are even and real-valued, hence so aretheir Fourier transforms (obtained via (cid:70) or (cid:70) − without difference).It is therefore suitable, in the scope of this subsection, to reserve thenotation (cid:83) ( R ) to real-valued Schwartz functions and to describe thedesired Fourier doublets [ S (cid:55)→ Σ] in terms of real-valued function al-gebras ( S, (cid:63) ) ≤ (cid:83) (cid:0) R , (cid:63) (cid:1) and (Σ , · ) ≤ (cid:0) (cid:83) ( R ) , · (cid:1) . One may of coursesubsequently employ complexification if complex-valued functions aredesired.We shall use the generator s on the spectral side. In other words, wedefine the spectral space as Σ = R [ s ] + , the nonunital algebra of poly-nomials having positive degree. This time we shall follow the expedientcustom of writing signals as functions in x and spectra in ξ . Thus wehave s ( ξ ) ∈ Σ as generator.For writing out its Fourier transform, it is useful to introduce acertain polynomial sequence σ n that somehow plays the role of theHermite polynomials in the Gaussian case. Using binomial coefficientsor falling factorials, we can write them as σ n ( t ) := (2 i ) k (cid:0) ( n − / − itn (cid:1) = (2 i ) n n ! ( n − − it ) n for n ≥ . It is easy to see that n ! σ n ( t ) is given by (cid:81) kj =1 (cid:0) t + (2 j + 1) (cid:1) for n = 2 k and by n t (cid:81) kj =1 ( t + j ) for n = 2 k + 1 , so these are in factreal polynomials (which are odd/even exactly when n is odd/even, justas with the Hermite polynomials).The Fourier transform of arbitrary positive powers of the hyperbolicsecant is a polynomial multiple of either the hyperbolic secant or hyper-bolic cosecant , according as the exponent is odd or even, respectively.Setting c ( t ) := csch( πt ) , we have for k ≥ the Fourier transforms(100) (cid:40) (cid:70) (cid:0) s ( ξ ) k +1 (cid:1) = σ k ( x ) s ( x ) ∈ R [ x ] s ( x ) , (cid:70) (cid:0) s ( ξ ) k +2 (cid:1) = σ k +1 ( x ) c ( x ) ∈ R [ x ] x c ( x ) , N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 119 which are all real-valued even functions, as expected.It is thus clear that the image of the spectral space under (cid:70) , andunder (cid:70) − alike, is the R [ x ] -submodule S generated by s, xc ∈ (cid:83) ( R ) .While this yields the signal space as a real vector space, we mustyet identify the convolution product on it. To this end, note that S = R [ x ] s ⊕ R [ x ] x c , so every signal may uniquely be written in theform u ( x ) = p ( x ) s ( x ) + q ( x ) x c ( x ) with polynomials p ( t ) , q ( t ) ∈ R [ t ] .Incidentally, this shows that S ∼ = R [ x ] as real vector spaces (formallysetting s = c = 1 ). Since the σ n ( t ) are a basis of R [ t ] , there are uniquecoefficients c , c , c , . . . ∈ R , essentially given by the Stirling partitionnumbers [92, (4.1.3)], such that u ( x ) = deg( p ) (cid:88) k =0 c k σ k ( x ) s ( x ) + deg( q ) (cid:88) k =0 c k +1 σ k +1 ( x ) c ( x ) . Since convolution is bilinear on (cid:83) ( R ) , it suffices to compute all productsbetween σ k ( x ) s ( x ) = (cid:70) (cid:0) s ( ξ ) k +1 (cid:1) and σ k +1 ( x ) c ( x ) = (cid:70) (cid:0) s ( ξ ) k +2 (cid:1) ;these are given by (cid:70) (cid:0) s ( ξ ) m (cid:1) (cid:63) (cid:70) (cid:0) s ( ξ ) n (cid:1) = (cid:70) (cid:0) s ( ξ ) m + n (cid:1) ∈ S by (100)since (cid:70) is an isomorphism between (cid:0) (cid:83) ( R ) , (cid:63) (cid:1) and (cid:0) (cid:83) ( R ) , · (cid:1) . Remark 86.
Admitting the full Heisenberg group H ( (cid:36) ) = TR (cid:111) R would commit us to adding all translates s b ( t ) := s ( t − b ) for b ∈ R ,which could be restricted for reasons of constructivity to rational valuesof b as in the case of the rational Gaussian singlet in §4.3.For seeing that the functions s b are algebraically independent over R ,assume a polynomial relation r ( t ) = (cid:80) α ∈ N m c α s b ( t ) α · · · s b m ( t ) α m ofminimal total degree, for translates by b , . . . , b m ∈ R and coefficients c α ∈ R . Taking the limit | t | → ∞ shows that we must have a = 0 since all powers of s b vanish at ∞ . But then we may factor the relationas s b ( t )˜ r ( t ) = 0 for some translate s b and a similar relation ˜ r ( t ) ofsmaller total degree. This is a contradiction since we may divide by s b > . Since the s b are algebraically independent, the Heisenberg H ( (cid:36) ) -algebra generated by s may be viewed as the tensor productof C ( R ) with the algebra R [ s b | b ∈ R ] + of multivariate polynomialsof positive degree. Here C ( R ) is the C -linear span of the oscillatingexponentials e α , as we have also used it in §4.1 for the Gaussians.The problem is that taking Fourier transforms of products suchas s b s b (cid:48) , or equivalently convolving s b and s b (cid:48) , leads to factors involv-ing beta functions whose algebraic treatment might be cumbersome.Further investigation on this topic would be needed. (cid:125)
20 MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ We can nevertheless carry the Fourier doublet [ S (cid:55)→ Σ] one stepfurther, achieving closure under the action of the Weyl algebra . Tothis end, we go back to our usual interpretation of (cid:83) ( R ) as the alge-bra of complex-valued Schwartz functions on which A ( R ) acts natu-rally. Defining the hyperbolic tangent in the form h ( t ) := tanh( πt ) , wehave s (cid:48) = − π hs and h (cid:48) = π s so that C [ h ][ s ] + is a differential subalge-bra of (cid:83) ( R ) . Note that C [ h ][ s ] + consists of all polynomials in t and s that have positive degree in s , but we prefer to see this as enlarging our(complexified) previous spectral space C Σ = C [ s ] + by extending thescalar ring from C to C [ h ] . Since h = 1 − s , we need only expand C Σ by the functions hs n .It is easy to find their Fourier transforms . Using ( s n ) (cid:48) = − nπ hs n and (42), we find (cid:70) ∨ ( hs n ) = − i k +1 x (cid:70) ( s n ) , where (100) may be employed for computing the Fourier transformof s n . Note that this time we must explicitly use the backward Fouriertransform (cid:70) ∨ since the functions hs n are odd so that their imagesunder (cid:70) ∧ differ from those under (cid:70) ∨ by a sign. By the same to-ken, the corresponding (forward or backward) Fourier transforms areimaginary- rather than real-valuede functions. On the signal side, the(complexified) space C S = C [ x ] s ⊕ C [ x ] x c gains the “missing” com-ponents C [ x ] x s ⊕ C [ x ] + c .For obtaining closure under the Weyl action , the space C [ h ][ s ] + mustbe extended by the polynomials C [ x ] ; thus we set Σ (cid:48) := C [ x, h ][ s ] + for the enlarged spectral space. Its image under (cid:70) ∨ or, equivalently,under (cid:70) ∧ is then enlarged from C [ x ] s ⊕ C [ x ] xc by adding in all deriva-tives. We call it S (cid:48) ⊂ (cid:83) ( R ) , but unfortunately its explicit characteriza-tion appears to be cumbersome and shall not be given here. We preferto proceed in a more roundabout way, enlarging S (cid:48) by meromorphicfunctions outside of (cid:83) ( R ) .Iterated derivatives of s and c may be computed using so-called de-rivative polynomials [22]. We shall not need their explicit form here;it suffices to know that the m -th derivative of s is given by s times acertain m -th degree polynomial in h while the corresponding derivativeof c is c times an m -th degree polynomial in h − . Clearly, all derivativesof xc are then given by similar expressions. Writing (cid:72) := C [ h, h − ][ x ] for the new coefficient ring, we may thus define S (cid:48)(cid:48) := (cid:72) s ⊕ (cid:72) c . Itis easy to see that S (cid:48)(cid:48) is a D -module (for D = A ( C ) , that is) con-taining S (cid:48) . Provided the elements of S (cid:48) are identified in the form w · s N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 121 or w · xc for a Weyl actor w ∈ A ( C ) , is is straightforward to com-pute their Fourier transforms via (42) and (100). Let us summarizeour results. Proposition 87.
The Fourier doublets [ S (cid:55)→ Σ] ≤ [ S (cid:48) (cid:55)→ Σ (cid:48) ] areregular subdoublets of [ (cid:83) ( R ) (cid:55)→ (cid:83) ( R )] qua plain doublet. Moreover, thecompatible Weyl action on the latter restricts to [ S (cid:48) (cid:55)→ Σ (cid:48) ] . Remark 88.
It would be significantly more ambititious to study theFourier singlet generated by s in (cid:83) ( R ) , even for the trivial Heisenberggroup as used above. We shall leave this as a challenge for futureinvestigations.We might consider (cid:65) := C [ x ][ c, s | c s − c − s ] , which is an algebraand a D -module. Then we might adjoin h and h − to (cid:65) , subject tothe further relations h = 1 − s , h − = 1 + c and h − − h = sc .Computations in Mathematica suggest that most (or all?) Fouriertransforms of such functions can be determined explicitly. But certainsimplifications such as tanh( πt/
2) = h − ( t ) − c ( t ) and sech ( πt/
2) = s ( t ) c ( t ) − c ( t ) + s ( t ) may be needed. Moroever, one may need largerfunction spaces (perhaps not quite distributions) for justifying (cid:70) ( h ) = ic and (cid:70) ( h − ) = ih − , but maybe one should not adjoin these functionsthemselves (only their products with “moderating” functions such as s ). (cid:125) Acknowledgments
We are very grateful to Heinrich Rolletschek for his helpful andfriendly advice on number-theoretic issues.
References [1] Ralph Abraham, Jerrold E. Marsden, and Tudor Raţiu.
Manifolds, tensoranalysis, and applications , volume 2 of
Global Analysis Pure and Applied:Series B . Addison-Wesley, Reading (Massachussettes), 1983. 19[2] F. Abtahi, R. Nasr-Isfahani, and A. Rejali. Convolution on L p -spaces of alocally compact group. Mathematica Slovaca , 63(2):291–298, 2013. 68[3] S. S. Akbarov. Fourier transform of invariant differential operators on alocally-compact abelian group.
Mathematical Notes , 56(2):852–855, Aug 1994.75[4] S S Akbarov. Construction of the cotangent bundle of a locally compact group.
Izvestiya: Mathematics , 59(3):445–470, 1995. 67[5] S S Akbarov. Differential geometry and quantization on a locally compactgroup.
Izvestiya: Mathematics , 59(2):271–286, 1995. 67[6] S S Akbarov. Smooth structure and differential operators on a locally com-pact group.
Izvestiya: Mathematics , 59(1):1–44, 1995. 5, 70, 73, 74, 75[7] Saban Alaca and Kenneth S. Williams.
Introductory Algebraic Number The-ory . Cambridge University Press, 2003. 101
22 MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ [8] Sergio Ardanza-Trevijano, María-Jesús Chasco, and Xabier Domínguez. Therole of real characters in the pontryagin duality of topological abelian groups. Journal of Lie Theory , 18:193–203, 2008. 75[9] G.B. Arfken, H.J. Weber, and F.E. Harris.
Mathematical Methods for Physi-cists: A Comprehensive Guide . Elsevier Science, 2013. 112, 115[10] Matthias Aschenbrenner and Lou van den Dries. Asymptotic differential alge-bra. In
Analyzable functions and applications , volume 373 of
Contemp. Math. ,pages 49–85. Amer. Math. Soc., Providence, RI, 2005. 85, 86[11] Franz Baader and Tobias Nipkow.
Term rewriting and all that . CambridgeUniversity Press, Cambridge, 1998. 107[12] Leo Bachmair and Harald Ganzinger. Buchberger’s Algorithm: A Constraint-Based Completion Procedure. In
Constraints in Computational Logics - FirstInternational Conference (CCL’94) , volume 845 of
Lecture Notes in ComputerScience , pages 285–301. Springer, 1994. 108[13] Alan Baker.
Transcendental Number Theory . Cambridge Univ. Press, 1975.99[14] Michael Barr and Charles Wells, editors.
Category Theory for ComputingScience, 2Nd Ed.
Prentice Hall International (UK) Ltd., Hertfordshire, UK,UK, 1995. 23[15] Thomas Becker and Volker Weispfenning.
Gröbner bases , volume 141 of
Grad-uate Texts in Mathematics . Springer, New York, 1993. A computational ap-proach to commutative algebra, In cooperation with Heinz Kredel. 107[16] R.J. Beerends, H.G. ter Morsche, J.G. van den Berg, and E.M. van de Vrie.
Fourier and Laplace Transforms . Fourier and Laplace Transforms. CambridgeUniversity Press, 2003. 43, 50, 52[17] Chal Benson and Gail Ratcliff. Gelfand pairs associated with finite heisen-berg groups. In Palle E.T. Jorgensen, Kathy D. Merrill, and Judith A. Packer,editors,
Representations, Wavelets, and Frames. A Celebration of the Mathe-matical Work of Lawrence W. Baggett . Birkhäuser, 2008. 14[18] E. Binz and S. Pods.
The Geometry of Heisenberg Groups: With Applicationsin Signal Theory, Optics, Quantization, and Field Quantization . Mathemat-ical surveys and monographs. American Mathematical Society, 2008. 17[19] J.E. Björk.
Rings of Differential Opreators . North-Holland Publishing Co.,1979. 108[20] Marco Bonatto and Dikran Dikranjan. Generalized heisenberg groups andself-duality. Preprint on arXiv:1611.02685, September 2017. 7, 9[21] Nicolas Bourbaki.
Lie Groups and Lie Algebras: Chapters 1-3 . Bourbaki,Nicolas: Elements of mathematics. Springer-Verlag, 1989. 78[22] Khristo N. Boyadzhiev. Derivative polynomials for tanh, tan, sech and sec inexplicit form.
Fibonacci Quarterly , 45(4):291–303, 2007. 120[23] John P. Boyd. The fourier transform of the quartic gaussian exp(-ax4): Hy-pergeometric functions, power series, steepest descent asymptotics and hyper-asymptotics and extensions to exp(-ax2n).
Applied Mathematics and Compu-tation , 241:75–87, 2014. 110, 111[24] Ronald Bracewell.
The Fourier Transform and its Applications . McGraw-Hill,1986. 43, 46, 50, 56, 67, 78, 110, 112[25] François Bruhat. Sur les représentations induites des groupes de Lie.
Bull.Soc. Math. France , 89:43–75, 1961. 69, 72, 74
N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 123 [26] W. Bruns and J. Gubeladze.
Polytopes, Rings, and K-Theory . Springer Mono-graphs in Mathematics. Springer New York, 2009. 100[27] Bruno Buchberger and Rüdiger Loos. Algebraic simplification. In BrunoBuchberger, G.E. Collins, and Rüdiger Loos, editors,
Computer Algebra -Symbolic and Algebraic Computation , pages 11–43. Copyrigth: Springer Ver-lag, Vienna - New York, 1982. 12, 103, 104, 109[28] Bruno Buchberger and Markus Rosenkranz. Transforming problems fromanalysis to algebra: a case study in linear boundary problems.
J. Symb. Com-put. , 47(6):589–609, 2012. 2, 79[29] P. M. Cohn.
Basic algebra: Groups, Rings and Fields . Springer, London, 2003.92, 93, 101, 102[30] P. M. Cohn.
Further algebra and applications . Springer-Verlag London Ltd.,London, 2003. 27, 29[31] S. C. Coutinho.
A primer of algebraic D -modules , volume 33 of LondonMathematical Society Student Texts . Cambridge University Press, Cambridge,1995. 77, 78[32] M.A. de Gosson.
Symplectic Geometry and Quantum Mechanics . OperatorTheory: Advances and Applications. Birkhäuser Basel, 2006. 54[33] J. E. Diem and F. B. Wright. Real characters and the radical of an abeliangroup.
Transactions of the American Mathematical Society , 129(3):517–529,1967. 75[34] J. Dieudonné.
Infinitesimal Calculus . Kershaw, 1973. 86[35] David Eisenbud.
Commutative algebra: With a view toward algebraic geom-etry , volume 150 of
Graduate Texts in Mathematics . Springer-Verlag, NewYork, 1995. 87[36] Philippe Flajolet and Robert Sedgewick.
Analytic combinatorics . CambridgeUniversity Press, Cambridge, 2009. 109[37] G.B. Folland.
A Course in Abstract Harmonic Analysis . Studies in AdvancedMathematics. Taylor & Francis, 1994. 15, 43, 46, 47, 48, 58, 65, 68[38] G.B. Folland.
Harmonic Analysis in Phase Space . Annals of MathematicsStudies. Princeton University Press, 2016. 17, 18, 21, 54, 67[39] Xing Gao, Li Guo, and Markus Rosenkranz. On rings of differential Rota-Baxter operators.
Internat. J. Algebra Comput. , 28(1):1–36, 2017. 79, 80, 81[40] Katarzyna Gorska and Karol A. Penson. Exact and explicit evaluation ofBrezin-Hikami kernels.
Nucl. Phys. , B872:333–347, 2013. 110, 111[41] C. Grunspan and J. van der van der Hoeven. Effective asymptotic analysisfor finance. Technical report, HAL, 2017. 86[42] Li Guo, Georg Regensburger, and Markus Rosenkranz. On integro-differentialalgebras.
J. Pure Appl. Algebra , 218(3):456–473, 2014. 79, 80[43] B.C. Hall.
Quantum Theory for Mathematicians . Graduate Texts in Mathe-matics. Springer New York, 2013. 19[44] E. Hewitt and K.A. Ross.
Abstract Harmonic Analysis: Structure of topo-logical groups, integration theory, group representations . Abstract HarmonicAnalysis. Springer-Verlag, second edition, 1994. 75, 76[45] E. Hewitt and K.A. Ross.
Abstract Harmonic Analysis: Volume II: Struc-ture and Analysis for Compact Groups Analysis on Locally Compact AbelianGroups . Grundlehren der mathematischen Wissenschaften. Springer BerlinHeidelberg, 1997. 41
24 MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ [46] Peter John Hilton and Urs Stammbach. A course in homological algebra .Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics,Vol. 4. 88[47] Roger Howe. On the role of the heisenberg group in harmonic analysis.
Bul-letin of the AMS , 3(2):821–843, September 1980. 66[48] Roger Howe. Quantum mechanics and partial differential equations.
Journalof Functional Analysis , 38(2):188 – 254, 1980. 66[49] K.B. Howell.
Principles of Fourier Analysis, Second Edition . Textbooks inMathematics. CRC Press, 2016. 52[50] Thomas W. Hungerford.
Algebra , volume 73 of
Graduate Texts in Mathemat-ics . Springer, New York, 1980. Reprint of the 1974 original. Call Number10169 at IBM Linz. 100[51] Mads Sielemann Jakobsen.
Gabor frames on locally compact abelian groupsand related topics . PhD thesis, Department of Applied Mathematics and Com-puter Science, Technical University of Denmark, September 30 2016. 55[52] K. Kato, N. Kurokawa, and T. Saito.
Number Theory 2: Introduction to ClassField Theory . Translations of mathematical monographs. American Mathe-matical Society, 2011. 99[53] Andrei V. Kelarev.
Ring Constructions and Applications . World Sci. Publ.,2002. 29[54] Vladimir V. Kisil. Plain mechanics: Classical and quantum.
J. of NaturalGeometry , 9(1):1–14, 1996. arXiv:funct-an/9405002. 66[55] Vladimir V. Kisil. Quantum and classic brackets.
Int. J. Theor. Phys. ,41(1):63–77, 2002. arXiv:math-ph/0007030. 66[56] Vladimir V. Kisil. Observables and states in p -mechanics. Advances in Math-ematics Research , V:101–136, 2003. arXiv:quant-ph/0304023. 66[57] Vladimir V. Kisil. p -mechanics as a physical theory: An introduction. J. Phys.A , 37(1):183–204, arXiv:quant-ph/0212101 2004. 66[58] Vladimir V. Kisil. Erlangen program at large. Lectures notes for a Postgrad-uate Course, August 3 2018. 66[59] Anthony W. Knapp.
Basic real analysis . Cornerstones. Birkhäuser BostonInc., Boston, MA, 2005. 50, 59[60] E.R. Kolchin.
Differential algebra and algebraic groups , volume 54 of
Pureand Applied Mathematics . Academic Press, New York-London, 1973. 42[61] Günter Landsmann and Markus Rosenkranz. Heisenberg groups via algebra.Preprint., 2019. 3[62] Serge Lang.
Algebra , volume 211 of
Graduate Texts in Mathematics . Springer-Verlag, third edition, 2002. 14, 45[63] R. Lakshmi Lavanya. A characterisation of the Fourier transform on theSchwartz-Bruhat space of locally compact abelian groups. Preprint onarXiv:1604.07533, April 2016. 43[64] Robert A. Leslie. How not to repeatedly differentiate a reciprocal.
Am. Math.Monthly , 98(10):732–735, 1991. 116[65] P. Libermann and C.M. Marle.
Symplectic Geometry and Analytical Mechan-ics . Mathematics and Its Applications. Springer Netherlands, 2012. 17[66] Gunar E. Liepins. A paley-wiener theorem for locally compact abelian groups.
Transactions of the American Mathematical Society , 222:193–210, 1976. 75,76, 81, 82, 83
N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 125 [67] L.H. Loomis.
Introduction to Abstract Harmonic Analysis . Dover Books onMathematics. Dover Publications, 2013. 46[68] Franz Luef and Yuri I. Manin. Quantum theta functions and gabor framesfor modulation spaces.
Letters in Mathematical Physics , 88(1):131, Feb 2009.7[69] George W. Mackey. The laplace transform for locally compact abelian groups.
Proceedings of the National Academy of Sciences of the United States of Amer-ica , 34(4):156–162, 1948. 75, 81[70] Saunders MacLane.
Homology . Classics in Mathematics. Springer-Verlag,Berlin, 1995. Reprint of the 1975 edition. 29[71] Saunders MacLane.
Categories for the Working Mathematician . Springer,Berlin, second edition, 1998. 12, 33[72] Y.I. Manin and A.A. Panchishkin.
Introduction to Modern Number Theory:Fundamental Problems, Ideas and Theories . Encyclopaedia of MathematicalSciences. Springer Berlin Heidelberg, 2006. 104, 111[73] Jerrold E. Marsden and Tudor S. Ratiu.
Introduction to mechanics and sym-metry , volume 17 of
Texts in Applied Mathematics . Springer-Verlag, NewYork, 1994. A basic exposition of classical mechanical systems. 19[74] Jan Mikusiński.
Operational calculus , volume 8 of
International Series ofMonographs on Pure and Applied Mathematics . Pergamon Press, New York,1959. 84[75] S.A. Morris.
Pontryagin Duality and the Structure of Locally Compact AbelianGroups . Cambridge University Press, 1977. 12, 70[76] Martin Moskowitz. Homological algebra in locally compact abelian groups.
Transactions of the American Mathematical Society , 127(3):361–404, 1967.55, 69[77] David Mumford.
Tata lectures on theta. III . Modern Birkhäuser Classics.Birkhäuser Boston, Inc., Boston, MA, 2007. With collaboration of MadhavNori and Peter Norman, Reprint of the 1991 original. 67[78] L. Nachbin.
The Haar Integral . Robert E. Krieger, 1976. 49, 55, 56[79] Peter J. Nicholson. Algebraic theory of finite fourier transforms.
Journal ofComputer and System Sciences , 5(5):524 – 547, 1971. 15, 54[80] Toshinori Oaku, Yoshinao Shiraki, and Nobuki Takayama. Algebraic algo-rithms for d -modules and numerical analysis. International Journal of Com-puter Mathematics - IJCM , 03 2003. 109[81] A.V. Oppenheim and R.W. Schafer.
Discrete-time Signal Processing .Prentice-Hall signal processing series. Pearson, 2010. 53[82] A.V. Oppenheim and A.S. Willsky.
Signals and Systems . Always learning.Pearson Education Limited, 2013. 52, 65[83] M. Scott Osborne. On the schwartz-bruhat space and the paley-wiener the-orem for locally compact abelian groups.
Journal of Functional Analysis ,19:40–49, 1975. 70, 71, 76[84] Amritanshu Parasad and M. K. Vemuri. Decomposition of phase space andclassification of heisenberg groups. Preprint on arXiv:0806.4064., June 2008.7[85] Amritanshu Prasad, Ilya Shapiro, and M. K. Vemuri. Locally compact abeliangroups with symplectic self-duality.
Adv. Math. , 225(5):2429–2454, 2010. 7
26 MARKUS ROSENKRANZ AND GÜNTER LANDSMANN ∗ [86] Danuta Przeworska-Rolewicz. Algebraic analysis . PWN—Polish ScientificPublishers, Warsaw, 1988. 112[87] Marc A. Rieffel. Projective modules over higher-dimensional non-commutative tori.
Can. J. Math. , XL(2):257–338, 1988. 21[88] Robert H. Risch. The problem of integration in finite terms.
Trans. Amer.Math. Soc. , 139:167–189, 1969. 3[89] Joseph Fels Ritt.
Differential algebra . Dover, New York, 1966. 42[90] M.J. Roberts.
Signals and Systems: Analysis using Transform Methods andMATLAB . McGraw Hill, 2012. 51, 65[91] D. W. Roeder. Functorial characterization of pontryagin duality.
Trans.Amer. Math. Soc. , 154:151–175, 1971. 69[92] Steven Roman.
The Umbral Calculus . Dover Publications Inc., 2005. 113, 114,115, 117, 119[93] Markus Rosenkranz. A new symbolic method for solving linear two-pointboundary value problems on the level of operators.
J. Symb. Comput. ,39(2):171–199, 2005. 2[94] Markus Rosenkranz, Xing Gao, and Li Guo. An algebraic study of multivari-able integration and linear substitution.
J. Algebra Appl. , 18(11):1950207,November 2019. 2, 79[95] Markus Rosenkranz and Nalina Phisanbut. A symbolic approach to boundaryproblems for linear partial differential equations. In V.P. Gerdt, W. Koepf,E.W. Mayr, and E.V. Vorozhtsov, editors,
Proceedings of the 15th Int. Work-shop on Computer Algebra in Scientific Computing , volume 8136 of
LectureNotes in Computer Science , pages 301–314. Springer, 2013. 3[96] Markus Rosenkranz and Georg Regensburger. Solving and factoring boundaryproblems for linear ordinary differential equations in differential algebras.
J.Symb. Comput. , 43(8):515–544, 2008. 2, 79[97] Markus Rosenkranz, Georg Regensburger, Loredana Tec, and Bruno Buch-berger. A symbolic framework for operations on linear boundary problems.In V.P. Gerdt, E.W. Mayr, and E.V. Vorozhtsov, editors,
Proceedings of the11th Int. Workshop on Computer Algebra in Scientific Computing , volume5743 of
Lecture Notes in Computer Science , pages 269–283. Springer, 2009. 2[98] Markus Rosenkranz, Georg Regensburger, Loredana Tec, and Bruno Buch-berger. Symbolic analysis for boundary problems: From rewriting toparametrized Gröbner bases. In Ulrich Langer and Peter Paule, editors,
Nu-merical and Symbolic Scientific Computing: Progress and Prospects , pages273–331. Springer Vienna, 2012. 2[99] Markus Rosenkranz and Nitin Serwa. An integro-differential structure forDirac distributions.
J. Symb. Comput. , 92:156–189, May–June 2019. 48, 79,83[100] W. Rudin.
Fourier Analysis on Groups . Dover Books on Mathematics. DoverPublications, 2017. 12, 15, 45, 46, 47, 48, 53, 57, 58, 60, 65, 76[101] Walter Rudin, Yitzhak Katznelson, Jean-Pierre Kahane, and Henry Helson.The functions which operate on fourier transforms.
Acta Mathematica , 102(1-2):135–157, 1959. 57[102] C. Sabbah.
Isomonodromic Deformations and Frobenius Manifolds: An In-troduction . Universitext. Springer London, 2007. 77, 78[103] Laurent Schwartz.
Théorie des distributions 1–2 . Hermann, 1951. 69
N ALGEBRAIC APPROACH TO FOURIER TRANSFORMATION 127 [104] S.W. Smith.
The Scientist and Engineer’s Guide to Digital Signal Processing .California Technical Pub., 1997. 52, 53, 65[105] E. Stade.
Fourier Analysis . Pure and Applied Mathematics: A Wiley Seriesof Texts, Monographs and Tracts. Wiley, 2011. 50, 57, 58, 59, 112[106] Ivar Stakgold.
Green’s functions and boundary value problems . Wiley, NewYork, 1979. 79, 80, 81[107] Stephen M. Stigler. A modest proposal: A new standard for the normal.
TheAmerican Statistician , 36(2):137–138, 1982. 67[108] Robert Strichartz.
A Guide to Distribution Theory and Fourier Transforms .CRC Press, 1994. 46, 69, 78, 80, 83, 112[109] Bernard Teissier. Valuations, deformations, and toric geometry. In F.V.Kuhlmann, S. Kuhlmann, and M. Marshall, editors,
Valuation theory andits applications, Vol. II , volume 33 of
Fields Institute Communications , pages361–459, Providence, RI, 2003. Amer. Math. Soc. 93[110] B. Thaller.
Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena . Springer NewYork, 2000. 108, 112[111] Marius van der Put and Michael F. Singer.
Galois theory of linear differentialequations , volume 328 of
Grundlehren der Mathematischen Wissenschaften .Springer, Berlin, 2003. 3[112] B. L. van der Waerden.
Algebra I . Springer, New York, ninth edition, 2003.103[113] M. K. Vemuri. Realizations of the canonical representation.
Proc. IndianAcad. Sci. Math. Sci. , 118(1):115–131, 2008. 7[114] E. Völkl, L.F. Allard, and D.C. Joy.
Introduction to Electron Holography .Kluwer Academic/Plenum Publishers, 1999. 46[115] S. Walters. Periodic integral transforms and C ∗ algebras. C. R. Math. Rep.Acad. Sci. Canada , 26(2):55–61, 2004. 21[116] R.B. Warfield.
Nilpotent groups . Lecture notes in mathematics. Springer,1976. 8[117] A. Wawrzyńczyk. On tempered distributions and Bochner-Schwartz theo-rem on arbitrary lolocal compact abelian groups.
Colloquium Mathematicum ,XIX(2):305–318, 1968. 70, 72, 74[118] Charles Weibel.
An Introduction to Homological Algebra . Cambridge Univ.Press, 1994. 29[119] André Weil. Sur certains groupes d’opérateurs unitaires.
Acta mathematica ,111:143–211, 1964. 12, 69[120] K. Yosida.
Operational calculus , volume 55 of
Applied Mathematical Sciences .Springer-Verlag, New York, 1984. A theory of hyperfunctions. 84[121] Doron Zeilberger. A holonomic systems approach to special functions identi-ties.
J. Comput. Appl. Math. , 32(3):321–368, 1990. 108, 109
RISC, Johannes Kepler University, A-4040 Linz, Austria
E-mail address ::