AAffine Quantum Harmonic Analysis
Eirik Berge, Stine M. Berge, Franz Luef, Eirik Skrettingland
Abstract
We develop a quantum harmonic analysis framework for the affine group. This encap-sulates several examples in the literature such as affine localization operators, covariantintegral quantizations, and affine quadratic time-frequency representations. In the pro-cess, we develop a notion of admissibility for operators and extend well known results tothe operator setting. A major theme of the paper is the interaction between operatorconvolutions, affine Weyl quantization, and admissibility.
The affine group and the Heisenberg group play prominent roles in wavelet theory and Gaboranalysis, respectively. As is well-known, the representation theory of the Heisenberg groupis intrinsically linked to quantization on phase space R n . Similarly, the relation betweenquantization schemes on the affine group and its representation theory has received someattention and several schemes have been proposed, e.g. [18, 5, 21]. However, there are stillmany open questions awaiting a definite answer in the case of the affine group.As has been shown by two of the authors in [36], the theory of quantum harmonic analysison phase space introduced by Werner [46] provides a coherent framework for many aspectsof quantization and Gabor analysis associated with the Heisenberg group. Based on thisconnection, advances in the understanding of time-frequency analysis have been made [37,38, 39]. In this paper we aim to develop a variant of Werner’s quantum harmonic analysis in[46] for time-scale analysis. This is based on unitary representations of the affine group in asimilar way to the Schr¨odinger representation of the Heisenberg group being used in Werner’sframework. We will refer to this theory on the affine group as affine quantum harmonicanalysis . Affine Operator Convolutions
In Werner’s quantum harmonic analysis on phase space, a crucial component is extendingconvolutions to operators. Recall that the affine group Aff has the underlying set R × R + andgroup operation modeling composition of affine transformations. A key feature of this groupis that the left Haar measure a − dx da and the right Haar measure a − dx da are not equal,making the group non-unimodular . Both measures play a role in affine quantum harmonicanalysis, making the theory more involved than the case of the Heisenberg group. In additionto the standard function (right-)convolution on the affine group f ∗ Aff g ( x, a ) := (cid:90) Aff f ( y, b ) g (( x, a ) · ( y, b ) − ) dy dbb , a r X i v : . [ m a t h . F A ] F e b e introduce the following operator convolutions for operators on L ( R + ) := L ( R + , r − dr )in Section 3: • Let f ∈ L r (Aff) := L (Aff , a − dx da ) and let S be a trace-class operator on L ( R + ).We define the convolution f (cid:63) Aff S between f and S to be the operator on L ( R + ) givenby f (cid:63) Aff S := (cid:90) Aff f ( x, a ) U ( − x, a ) ∗ SU ( − x, a ) dx daa , where U is the unitary representation of Aff on L ( R + ) given by U ( x, a ) ψ ( r ) := e πixr ψ ( ar ) . • Let S be a trace-class operator and let T be a bounded operator on L ( R + ). Then wedefine the convolution S (cid:63) Aff T between S and T to be the function on Aff given by S (cid:63) Aff T ( x, a ) := tr( SU ( − x, a ) ∗ T U ( − x, a )) . The three convolutions are compatible in the following sense: Let f, g ∈ L r (Aff) and denoteby S a trace-class operator and by T a bounded operator, both on L ( R + ). Then( f (cid:63) Aff S ) (cid:63) Aff T = f ∗ Aff ( S (cid:63) Aff T ) ,f (cid:63) Aff ( g (cid:63) Aff S ) = ( f ∗ Aff g ) (cid:63) Aff S. Interplay Between Affine Weyl Quantization and Convolutions
Integral to the theory in this paper is the affine Wigner distribution and the associated affineWeyl quantization. The affine (cross-)Wigner distribution W ψ,φ Aff of φ, ψ ∈ L ( R + ) is thefunction on Aff given by W ψ,φ Aff ( x, a ) = (cid:90) ∞−∞ ψ (cid:18) aue u e u − (cid:19) φ (cid:18) aue u − (cid:19) e − πixu du. (1.1)Although at first glance the definition (1.1) might look unnatural, it can be motivated throughthe representation theory of the affine group as illustrated in [3]. We will elaborate onthis viewpoint in Section 5. One defines the affine Weyl quantization of f ∈ L r (Aff) := L (Aff , a − dx da ) as the operator A f given by (cid:104) A f φ, ψ (cid:105) L ( R + ) = (cid:68) f, W ψ,φ Aff (cid:69) L r (Aff) , for all φ, ψ ∈ L ( R + ) . We will explore the intimate relation between the convolutions and the affine Weyl quanti-zation. The following theorem, being a combination of Proposition 3.6 and Proposition 3.7,highlights this relation.
Theorem A.
Let f, g ∈ L r (Aff) , where g is additionally in L r (Aff) and square integrablewith respect to the left Haar measure. Then g (cid:63) Aff A f = A g ∗ Aff f ,A g (cid:63) Aff A f = f ∗ Aff ˇ g, where ˇ g ( x, a ) := g (( x, a ) − ) .
2e will exploit the previous theorem to define the affine Weyl quantization of tempereddistributions in Section 3.3. To do this rigorously, we will utilize a Schwartz space S (Aff) onthe affine group introduced in [5]. An important example we prove in Theorem 3.11 is theaffine Weyl quantization of the coordinate functions: Theorem B.
Let f x ( x, a ) := x and f a ( x, a ) := a be the coordinate functions on Aff . Theaffine Weyl quantizations A f x and A f a satisfy the commutation relation [ A f x , A f a ] = 12 πi A f a . This is, up to re-normalization, precisely the infinitesimal structure of the affine group.
We define affine parity operator P Aff as P Aff = A δ (0 , , where δ (0 , denotes the Dirac distribution at the identity element (0 , ∈ Aff. The followingresult, which will be rigorously stated in Section 3.5, builds on these definitions.
Theorem C.
The affine Weyl quantization A g of g ∈ S (Aff) can be written as A g = g (cid:63) Aff P Aff . Moreover, for φ, ψ such that φ ( e x ) , ψ ( e x ) ∈ S ( R ) , the affine Weyl symbol W ψ,φ Aff of the rank-one operator ψ ⊗ φ can be written as W ψ,φ Aff = ( ψ ⊗ φ ) (cid:63) Aff P Aff . Operator Admissibility
One of the key features of representations of non-unimodular groups is the concept of ad-missibility. Recall that the
Duflo-Moore operator D − corresponding to the representation U is the densely defined positive operator on L ( R + ) given by D − ψ ( r ) = r − / ψ ( r ). We willoften use that D − has a densely defined inverse given by D ψ ( r ) = r / ψ ( r ). A function ψ is said to be an admissible wavelet if ψ ∈ dom( D − ). It is well known [13] that admissiblewavelets satisfy the orthogonality relation (cid:90) Aff |(cid:104) φ, U ( − x, a ) ∗ ψ (cid:105) L ( R + ) | dx daa = (cid:107) φ (cid:107) L ( R + ) (cid:107)D − ψ (cid:107) L ( R + ) . (1.2)We extend the definition of admissibility to operators as follows: Definition.
Let S be a non-zero bounded operator on L ( R + ) that maps dom( D ) intodom( D − ). We say that S is admissible if the composition D − S D − is bounded on dom( D − )and extends to a trace-class operator D − S D − on L ( R + ).Note that the rank-one operator S = ψ ⊗ ψ for ψ ∈ L ( R + ) is admissible precisely when ψ is an admissible wavelet. In Section 4.2 we show that a large class of admissible operatorscan be constructed from Laguerre bases. The following result, which we prove in Corollary4.5, is motivated by [46, Lemma 3.1] and extends (1.2) to the operator setting.3 heorem D. Let S be an admissible operator on L ( R + ) . For any trace-class operator T on L ( R + ) , we have that T (cid:63) Aff S ∈ L r (Aff) with (cid:90) Aff T (cid:63) Aff S ( x, a ) dx daa = tr( T ) tr( D − S D − ) . Determining whether an operator is admissible or not can be a daunting task. We man-aged in Corollary 4.9 to find an elegant characterization in terms of operator convolutions ofadmissible operators that are additionally positive trace-class operators.
Theorem E.
Let S be a non-zero, positive trace-class operator. Then S is admissible if andonly if S (cid:63) Aff S ∈ L r (Aff) . The following result is derived in Section 4.4 and uses the affine Weyl quantization to showthat admissibility is an operator manifestation of the non-unimodularity of the affine group.
Theorem F. • Let f ∈ L r (Aff) be such that A f is a trace-class operator on L ( R + ) . Then tr( A f ) = (cid:90) Aff f ( x, a ) dx daa . • Let g ∈ L l (Aff) := L (Aff , a − dx da ) be such that A g is an admissible Hilbert-Schmidtoperator. Then tr (cid:0) D − A g D − (cid:1) = (cid:90) Aff g ( x, a ) dx daa . Relationship with Fourier Transforms
For completeness, we will also investigate how notions of Fourier transforms on the affinegroup fit into the theory, and use known results from abstract harmonic analysis to explorethe relationship between affine Weyl quantization and affine Fourier transforms. Recall thatthe integrated representation U ( f ) of f ∈ L l (Aff) is the operator on L ( R + ) given by U ( f ) ψ := (cid:90) Aff f ( x, a ) U ( x, a ) ψ dx daa , ψ ∈ L ( R + ) . We define the following operator Fourier transform in the affine setting.
Definition.
The affine Fourier-Wigner transform is the isometry F W sending a Hilbert-Schmidt operator on L ( R + ) to a function in L r (Aff) such that F − W ( f ) = U ( ˇ f ) ◦ D , f ∈ Im( F W ) ∩ L r (Aff) . The following result is proved in Proposition 5.7 and provides a connection between theaffine Fourier-Wigner transform and admissibility.
Theorem G.
Let A be a trace-class operator on L ( R + ) . The following are equivalent:1) F W ( A D − ) ∈ L r (Aff) . ) A D − extends from dom( D − ) to a Hilbert-Schmidt operator on L ( R + ) .3) A ∗ A is admissible. Another Fourier transform of interest is the (modified)
Fourier-Kirillov transform on theaffine group F KO given by( F KO f )( x, a ) = √ a (cid:90) R f (cid:18) vλ ( − u ) , e u (cid:19) e − πi ( xu + av ) du dv (cid:112) λ ( − u ) , f ∈ Im( F W ) . As in quantum harmonic analysis on phase space, we have that the affine Weyl quantizationis the composition of these Fourier transforms, see Proposition 5.8. In the affine setting wehave in general that F W ( f (cid:63) Aff S ) (cid:54) = F KO ( f ) F W ( S ) , F KO ( S (cid:63) Aff T ) (cid:54) = F W ( S ) F W ( T ) . This contrasts the analogous result in Werner’s original quantum harmonic analysis, see (5.6).In spite of this, not all properties typically associated with the Fourier transform are lost: InSection 5.2 we prove a quantum Bochner theorem in the affine setting.
Main Applications
In Section 6 we show that affine quantum harmonic analysis provides a conceptual frameworkfor the study of covariant integral quantizations and a version of the
Cohen class for the affinegroup. In addition, we show in Section 6.1 that if S is a rank-one operator, then the studyof operators f (cid:63) Aff S for functions f on Aff reduces to the study of time-scale localizationoperators [12].We have seen that affine Weyl quantization is given by f (cid:55)→ f (cid:63) Aff P Aff for f ∈ S (Aff).Inspired by this, we consider a whole class of quantization procedures: For any suitably niceoperator S on L ( R + ) we define a quantization procedure Γ S for functions f on Aff byΓ S ( f ) := f (cid:63) Aff S. This class of quantization procedures coincides with the covariant integral quantizations stud-ied by Gazeau and his collaborators motivated by applications in physics, see e.g. [21, 20, 19].Our results on affine quantum harmonic analysis are therefore also results on covariant inte-gral quantizations. In particular, the abstract notion of admissibility of an operator S impliesthat Γ S satisfies the simple property Γ S (1) = c · I L ( R + ) , where c is some constant, I L ( R + ) is the identity operator on L ( R + ), and 1( x, a ) = 1 for all( x, a ) ∈ Aff.As the name suggests, covariant integral quantizations Γ S satisfy a covariance property,namely U ( − x, a ) ∗ Γ S ( f ) U ( − x, a ) = Γ S ( R ( x,a ) − f ) , where R denotes right translations of functions on Aff. In Theorem 6.5 we point out that,by a known result on covariant positive operator valued measures [34, 9], this covariance as-sumption together with other mild assumptions completely characterize the covariant integral5uantizations. We have also seen that the affine cross-Wigner distribution is given for suffi-ciently nice ψ, φ by W ψ,φ Aff = ( ψ ⊗ φ ) (cid:63) Aff P Aff . Inspired by this and the description in [37] ofthe Cohen class of time-frequency distributions on R n , we make the following definition. Definition.
A bilinear map Q : L ( R + ) × L ( R + ) → L ∞ (Aff) belongs to the affine Cohenclass if Q = Q S for some operator S on L ( R + ), where Q S ( ψ, φ )( x, a ) := ( ψ ⊗ φ ) (cid:63) Aff S ( x, a ) = (cid:104) SU ( − x, a ) ψ, U ( − x, a ) φ (cid:105) L ( R + ) . We will show how properties of S (such as admissibility) influence properties of Q S , andobtain an abstract characterization of the affine Cohen class. Readers familiar with the Cohenclass on R n [11] will know that it is defined in terms of convolutions with the Wigner function.In the affine setting, we have the analogous result Q A f ( ψ, φ ) = W ψ,φ Aff ∗ Aff ˇ f . As we explain in Proposition 6.14, the affine class of quadratic time-frequency representations from [41] may be identified with a subclass of the affine Cohen class.
Structure of the Paper
In Section 2 we recall necessary background material for completeness. In particular, Section2.2 should serve as a brief reference for quantum harmonic analysis on phase space. We defineaffine operator convolution in Section 3.1 and show the relationship with the affine Weylquantization in Section 3.2. The affine parity operator will be introduced in Section 3.4, andits relationship to affine Weyl quantization will be explored in Section 3.5. We have dedicatedthe entirety of Section 4 to operator admissibility. Section 5 discusses affine Weyl quantizationfrom the viewpoint of representation theory. In particular, in Section 5.2 we derive a Bochnertype theorem for our setting. In Section 6.1 and Section 6.2 we relate our work to time-scalelocalization operators and covariant integral quantizations, respectively. Finally, in Section6.3 we define the affine Cohen class and derive some basic properties.
Notation:
Given a Hilbert space H we let L ( H ) denote the bounded operators on H . Thenotation S p ( H ) for 1 ≤ p < ∞ will be used for the Schatten-p class operators on H . We re-mark that S ( H )and S ( H ) are respectively the trace-class operators and the Hilbert-Schmidtoperators on H . The space S ∞ ( H ) is by definition L ( H ) for duality reasons. When theHilbert space in question is H = L ( R + ) := L ( R + , r − dr ), we will simplify the notation to S p := S p ( L ( R + )) for readability. We will denote by S ( R n ) the space of Schwartz functionson R n . For a function f on a group G , the function ˇ f is defined by ˇ f ( g ) = f ( g − ) for all g ∈ G . We begin by giving a brief introduction to the affine group and relevant constructions on it.The (reduced) affine group (Aff , · Aff ) is the Lie group whose underlying set is the upper half6lane Aff := R × R + := R × (0 , ∞ ), while the group operation is given by( x, a ) · Aff ( y, b ) := ( ay + x, ab ) , ( x, a ) , ( y, b ) ∈ Aff . We will often neglect the subscript in the group operation to improve readability. Moreover,we use the notation L ( x,a ) and R ( x,a ) to denote respectively the left-translation and right-translation by ( x, a ) ∈ Aff, acting on a function f : Aff → C by (cid:0) L ( x,a ) f (cid:1) ( y, b ) := f (( x, a ) − · Aff ( y, b )) , (cid:0) R ( x,a ) f (cid:1) ( y, b ) := f (( y, b ) · Aff ( x, a )) . Recall that the translation operator T x and the dilation operator D a are respectively given by T x f ( y ) := f ( y − x ) , D a f ( y ) := 1 √ a f (cid:16) ya (cid:17) , x, y ∈ R , a ∈ R + . (2.1)The following computation motivates the group operation on the affine group:( T x D a )( T y D b ) = T x T ay D a D b = T x + ay D ab . We can represent the affine group Aff and its Lie algebra aff in matrix formAff = (cid:26)(cid:18) a x (cid:19) (cid:12)(cid:12)(cid:12) a > , x ∈ R (cid:27) , aff = (cid:26)(cid:18) u v (cid:19) (cid:12)(cid:12)(cid:12) u, v ∈ R (cid:27) . The Lie algebra structure of aff is completely determined by (cid:20)(cid:18) (cid:19) , (cid:18) (cid:19)(cid:21) = (cid:18) (cid:19) . (2.2)An important feature of the affine group is that it is non-unimodular; the left and right Haarmeasures are respectively given by µ L ( x, a ) = dx daa , µ R ( x, a ) = dx daa . As such, the modular function on the affine group is given by ∆( x, a ) = a − . The affine groupis exponential, meaning that the exponential map exp : aff → Aff given byexp (cid:18) u v (cid:19) = (cid:18) e u v ( e u − u (cid:19) is a global diffeomorphism. Hence we can write the left and right Haar measures in exponentialcoordinates by the formulas µ L ( x, a ) = du dvλ ( u ) , µ R ( x, a ) = du dvλ ( − u ) , λ ( u ) := ue u e u − . (2.3)Throughout the paper, we will heavily use the spaces L pl (Aff) := L p (Aff , µ L ) and L pr (Aff) := L p (Aff , µ R ) for 1 ≤ p ≤ ∞ . 7 .2 Quantum Harmonic Analysis on the Heisenberg Group Before delving into quantum harmonic analysis on the affine group, it is advantageous toreview the Heisenberg setting, originally introduced by Werner [46]. There are three primaryconstructions that appear: (a) A quantization scheme, (b) an integrated representation, and(c) a way to define convolution that incorporates operators. We give a brief overview of thesethree constructions and refer the reader to [46, 23, 36] for more details.
The cross-Wigner distribution of φ, ψ ∈ L ( R n ) is given by W ( φ, ψ )( x, ω ) := (cid:90) R n φ (cid:18) x + t (cid:19) ψ (cid:18) x − t (cid:19) e − πiωt dt, ( x, ω ) ∈ R n . When φ = ψ we refer to W φ := W ( φ, φ ) as the Wigner distribution of φ ∈ L ( R n ). Thecross-Wigner distribution satisfies the orthogonality relation (cid:104) W ( φ , ψ ) , W ( φ , ψ ) (cid:105) L ( R n ) = (cid:104) φ , φ (cid:105) L ( R n ) (cid:104) ψ , ψ (cid:105) L ( R n ) , φ , φ , ψ , ψ ∈ L ( R n ) . Moreover, the Wigner distribution satisfies the marginal properties (cid:90) R n W φ ( x, ω ) dω = | φ ( x ) | , (cid:90) R n W φ ( x, ω ) dx = | ˆ φ ( x ) | , for φ ∈ S ( R n ).Our primary interest in the cross-Wigner distribution stems from the following connection:For each f ∈ L ( R n ) we define the operator L f : L ( R n ) → L ( R n ) by the formula (cid:104) L f φ, ψ (cid:105) L ( R n ) = (cid:104) f, W ( ψ, φ ) (cid:105) L ( R n ) , φ, ψ ∈ L ( R n ) . Then L f is the Weyl quantization of f , see [23, Ch. 14] for details. It is a non-trivial fact,see [42], that the Weyl quantization gives a well-defined isomorphism between L ( R n ) and S ( L ( R n )), the space of Hilbert-Schmidt operators on L ( R n ). Recall that the Heisenberg group H n is the Lie group with underlying manifold R n × R n × R and with the group multiplication( x, ω, t ) · ( x (cid:48) , ω (cid:48) , t (cid:48) ) := (cid:18) x + x (cid:48) , ω + ω (cid:48) , t + t (cid:48) + 12 (cid:0) x (cid:48) ω − xω (cid:48) (cid:1)(cid:19) . The Heisenberg group is omnipresent in modern mathematics and theoretical physics, see [27].For a Hilbert space H we let U ( H ) denote the unitary operators on H . The most importantrepresentation of the Heisenberg group is the Schr¨odinger representation ρ : H n → U ( L ( R n ))given by ρ ( x, ω, t ) φ ( y ) := e πit e − πixω M ω T x φ ( y ) , T x is the n -dimensional analogue of the translation operator defined in (2.1) and M ω is the modulation operator given by M ω φ ( y ) := e πiωy φ ( y ) , φ ∈ L ( R n ) . The Schr¨odinger representation is both irreducible and unitary. Let us use the abbreviatednotation z := ( x, ω ) ∈ R n and π ( z ) = M ω T x . Ignoring the central variable t , we can considerthe integrated Schr¨odinger representation ρ : L ( R n ) → L ( L ( R n )) given by ρ ( f ) = (cid:90) R n f ( z ) e − πixω π ( z ) dz, (2.4)where L ( L ( R n )) denotes the bounded linear operators on L ( R n ). We remark that theintegral in (2.4) is defined weakly. It turns out, see [15, Thm. 1.30], that the integratedrepresentation ρ extends from L ( R n ) ∩ L ( R n ) to a unitary map ρ : L ( R n ) → S ( L ( R n )). Given a function f ∈ L ( R n ) and a trace-class operator S ∈ S ( L ( R n )), their convolutionis the trace-class operator on L ( R n ) defined by f (cid:63) S := (cid:90) R n f ( z ) π ( z ) Sπ ( z ) ∗ dz. The convolution f (cid:63) S satisfies the estimate (cid:107) f (cid:63) S (cid:107) S ≤ (cid:107) f (cid:107) L (cid:107) S (cid:107) S .One can also define the convolution between two operators: For two trace-class operators S, T ∈ S ( L ( R n )) we define their convolution to be the function on R n given by S (cid:63) T ( z ) := tr( Sπ ( z ) P T P π ( z ) ∗ ) , where P ψ ( t ) := ψ ( − t ) is the parity operator . The convolution S (cid:63) T satisfies the estimate (cid:107)
S (cid:63) T (cid:107) L ≤ (cid:107) S (cid:107) S (cid:107) T (cid:107) S , and the important integral relation [46, Lem. 3.1] (cid:90) R n S (cid:63) T ( z ) dz = tr( S ) tr( T ) . (2.5)To see the connection with the Wigner distribution, we note that the cross-Wigner distributionof ψ, φ ∈ L ( R n ) can be written as W ( ψ, φ ) = ψ ⊗ φ (cid:63) P, (2.6)where ψ ⊗ φ denotes the rank-one operator on L ( R n ) given by( ψ ⊗ φ )( ξ ) := (cid:104) ξ, φ (cid:105) L ( R n ) ψ for ξ ∈ L ( R n ) . Similarly, the Weyl quantization of f ∈ L ( R n ) may be expressed in terms of operatorconvolutions: L f = f (cid:63) P. (2.7)Hence convolution with the parity operator P gives a convenient way to represent the Wignerdistribution and the Weyl quantization. 9inally, there is a Fourier transform for operators: Given a trace-class operator S ∈S ( L ( R n )) we define the Fourier-Wigner transform F W ( S ) of S to be the function on R n given by F W ( S )( z ) := e iπxω tr( Sπ ( z ) ∗ ) , z ∈ R n . (2.8)The Fourier-Wigner transform extends to a unitary map F W : S ( L ( R n )) → L ( R n ), whereit turns out the to be inverse of the integrated Schr¨odinger representation given in (2.4). By[15, Prop. 2.5] it is related to the Weyl transform by the elegant formula f = F σ ( F W ( L f )) , (2.9)where F σ denotes the symplectic Fourier transform . We briefly describe affine Weyl quantization and how this gives rise to the affine Wignerdistribution. There is a unitary representation π of the affine group Aff on L ( R + , r − dr )given by U ( x, a ) ψ ( r ) := e πixr ψ ( ar ) = 1 √ a M x D a ψ ( r ) , ψ ∈ L ( R + , r − dr ) . (2.10)Since r − dr is the Haar measure on R + we will write L ( R + ) := L ( R + , r − dr ). Later wealso consider another measure on R + and will be more explicit when the situation requires it.To define the quantization scheme we will utilize the Stratonovich-Weyl operator on L ( R + ) given by Ω( x, a ) ψ ( r ) := a (cid:90) R e − πi ( xu + av ) U (cid:18) ve u λ ( u ) , e u (cid:19) ψ ( r ) du dv. (2.11)The following result was shown in [18] and provides us with an affine analogue of Weylquantization. Proposition 2.1 ([18]) . There is a norm-preserving isomorphism between L r (Aff) and thespace of Hilbert-Schmidt operators on L ( R + ) . The isomorphism sends f ∈ L r (Aff) to theoperator A f on L ( R + ) defined weakly by A f ψ ( r ) := (cid:90) ∞−∞ (cid:90) ∞ f ( x, a )Ω( x, a ) ψ ( r ) da dxa , ψ ∈ L ( R + ) . We will refer to the association f (cid:55)→ A f as affine Weyl quantization , while f is called the affine (Weyl) symbol of A f . To emphasize the correspondence between a Hilbert-Schmidtoperator A and its affine symbol f we use the notation f A := f . The affine Weyl symbol ofan operator A is explicitly given by f A ( x, a ) = (cid:90) ∞−∞ A K ( aλ ( u ) , aλ ( − u )) e − πixu du, (2.12)where A K : R + × R + → C is the integral kernel of A defined by Aψ ( r ) = (cid:90) ∞ A K ( r, s ) ψ ( s ) dss , ψ ∈ L ( R + ) .
10y taking the affine Weyl symbol of the rank-one operator ψ ⊗ φ on L ( R + ) given by ψ ⊗ φ ( ξ ) = (cid:104) ξ, φ (cid:105) L ( R + ) ψ for ψ, φ, ξ ∈ L ( R + ), we obtain the following definition. Definition 2.2.
For φ, ψ ∈ L ( R + ) we define the affine (cross-)Wigner distribution W ψ,φ Aff tobe the function on Aff given for ( x, a ) ∈ Aff by W ψ,φ Aff ( x, a ) := (cid:90) ∞−∞ ψ ( aλ ( u )) φ ( aλ ( − u )) e − πixu du = (cid:90) ∞−∞ ψ (cid:18) aue u e u − (cid:19) φ (cid:18) aue u − (cid:19) e − πixu du. When φ = ψ we refer to W ψ Aff := W ψ,ψ Aff as the affine Wigner distribution of ψ . The weakinterpretation of the integral defining A f means that we have the relation (cid:104) A f φ, ψ (cid:105) L ( R + ) = (cid:68) f, W ψ,φ Aff (cid:69) L r (Aff) , (2.13)for f ∈ L r (Aff) and φ, ψ ∈ L ( R + ). The affine Wigner distribution satisfies the orthogonalityrelation (cid:90) ∞−∞ (cid:90) ∞ W ψ ,ψ Aff ( x, a ) W φ ,φ Aff ( x, a ) da dxa = (cid:104) ψ , φ (cid:105) L ( R + ) (cid:104) ψ , φ (cid:105) L ( R + ) , (2.14)for ψ , ψ , φ , φ ∈ L ( R + ). Moreover, the affine Wigner distribution also satisfies the marginalproperty (cid:90) ∞−∞ W ψ Aff ( x, a ) dx = | ψ ( a ) | , ( x, a ) ∈ Aff , (2.15)for all rapidly decaying smooth functions ψ on R + . We remark that a rapidly decaying smoothfunction (also called a Schwartz function ) ψ : R + → C is by definition a smooth function suchthat x (cid:55)→ ψ ( e x ) is a rapidly decaying function on R . The space of all rapidly decaying smoothfunctions on R + will be denoted by S ( R + ). We will later also need the space S (cid:48) ( R + ) ofbounded, anti-linear functionals on S ( R + ) called the tempered distributions on R + . For moreinformation regarding the affine Wigner distribution the reader is referred to [5]. In this part we introduce operator convolutions in the affine setting. We show that this notionis intimately related to affine Weyl quantization in Section 3.2. In Section 3.4 we will introducethe affine Grossmann-Royer operator, which will be essential in Section 3.5 where we provethe main connection between the affine Weyl quantization and the operator convolutions inTheorem 3.21. 11 .1 Definitions and Basic Properties
We begin by defining operator convolutions in the affine setting and derive basic properties.Recall that the usual convolution on the affine group with respect to the right Haar measureis given by f ∗ Aff g ( x, a ) := (cid:90) Aff f ( y, b ) g (( x, a ) · ( y, b ) − ) dy dbb . Remark.
Other sources, e.g. [16], use the left Haar measure and define the convolution to be f ∗ Aff L g (( x, a )) := ˇ f ∗ Aff ˇ g (( x, a ) − ) , where ˇ f ( x, a ) := f (( x, a ) − ). We will mainly work with the right Haar measure, and ourdefinition ensures that (cid:107) f ∗ Aff g (cid:107) L r (Aff) ≤ (cid:107) f (cid:107) L r (Aff) (cid:107) g (cid:107) L r (Aff) . Additionally, we have that R ( x,a ) ( f ∗ Aff g ) = ( R ( x,a ) f ) ∗ Aff g. Definition 3.1.
Let f ∈ L r (Aff) and let S be a trace-class operator on L ( R + ). We definethe convolution f (cid:63) Aff S between f and S to be the operator on L ( R + ) given by f (cid:63) Aff S := (cid:90) Aff f ( x, a ) U ( − x, a ) ∗ SU ( − x, a ) dx daa , where U is the unitary representation given in (2.10). The integral is a convergent Bochnerintegral in the space of trace-class operators. Remark.
1. As we will see later, using U ( − x, a ) instead of U ( x, a ) in Definition 3.1 ensures that theconvolution is compatible with the following covariance property of the affine Wignerdistribution: W U ( − x,a ) φ,U ( − x,a ) ψ Aff ( y, b ) = W φ,ψ Aff (( y, b ) · ( x, a )) . (3.1)2. The notation (cid:63) has a different meaning in [18], where it is used to denote the so-calledMoyal product of two functions defined on Aff. Definition 3.2.
Let S be a trace-class operator and let T be a bounded operator on L ( R + ).Then we define the convolution S (cid:63) Aff T between S and T to be the function on Aff given by S (cid:63) Aff T ( x, a ) := tr( SU ( − x, a ) ∗ T U ( − x, a )) . Remark.
Recently, [10] defined another notion of convolution of trace-class operators. Un-like our definition, this convolution produces a new trace-class operator, with the aim ofinterpreting the trace-class operators as an analogue of the Fourier algebra.It is straightforward to check that if f is a positive function and S, T are positive operators,then f (cid:63) Aff S is a positive operator and S (cid:63) Aff T is a positive function. Moreover, we have theelementary estimate (cid:107) f (cid:63) Aff S (cid:107) S ≤ (cid:107) f (cid:107) L r (Aff) (cid:107) S (cid:107) S (3.2)and (cid:107) S (cid:63) Aff T (cid:107) L ∞ (Aff) ≤ (cid:107) S (cid:107) S (cid:107) T (cid:107) L ( L ( R + )) . (3.3)The following result is proved by a simple computation.12 emma 3.3. For ψ, φ ∈ L ( R + ) and S ∈ L ( L ( R + )) , we have ( ψ ⊗ φ ) (cid:63) Aff S ( x, a ) = (cid:104) SU ( − x, a ) ψ, U ( − x, a ) φ (cid:105) L ( R + ) . In particular, for η, ξ ∈ L ( R + ) we have ( ψ ⊗ φ ) (cid:63) Aff ( η ⊗ ξ )( x, a ) = (cid:104) ψ, U ( − x, a ) ∗ ξ (cid:105) L ( R + ) (cid:104) φ, U ( − x, a ) ∗ η (cid:105) L ( R + ) , and ( ψ ⊗ ψ ) (cid:63) Aff ( ξ ⊗ ξ )( x, a ) = |(cid:104) ψ, U ( − x, a ) ∗ ξ (cid:105) L ( R + ) | . A natural question to ask is whether the three different notions of convolution we haveintroduced are compatible. The following proposition gives an affirmative answer to thisquestion.
Proposition 3.4.
Let f, g ∈ L r (Aff) , S ∈ S , and let T be a bounded operator on L ( R + ) .Then we have the compatibility equations ( f (cid:63) Aff S ) (cid:63) Aff T = f ∗ Aff ( S (cid:63) Aff T ) ,f (cid:63) Aff ( g (cid:63) Aff S ) = ( f ∗ Aff g ) (cid:63) Aff S. Proof.
The first equality follows from the computation( f ∗ Aff ( S (cid:63) Aff T )) ( x, a ) = (cid:90) Aff f ( y, b ) tr( SU ( − y, b ) U ( − x, a ) ∗ T U ( − x, a ) U ( − y, b ) ∗ ) dy dbb = (cid:90) Aff f ( y, b ) tr( U ( − y, b ) ∗ SU ( − y, b ) U ( − x, a ) ∗ T U ( − x, a )) dy dbb = tr (cid:18)(cid:90) Aff f ( y, b ) U ( − y, b ) ∗ SU ( − y, b ) dy dbb U ( − x, a ) ∗ T U ( − x, a ) (cid:19) = (( f (cid:63) Aff S ) (cid:63) Aff T ) ( x, a ) . We are allowed to take the trace outside the integral since the second to last line is essentiallythe duality action of the bounded operator U ( − x, a ) ∗ T U ( − x, a ) on a convergent Bochnerintegral in the space of trace-class operators.For the second equality, we use change of variables and obtain( f ∗ Aff g ) (cid:63) Aff S = (cid:90) Aff (cid:90) Aff f ( x, a ) g (( z, c ) · ( x, a ) − ) U ( − z, c ) ∗ SU ( − z, c ) dx daa dz dcc = (cid:90) Aff (cid:90) Aff f ( x, a ) g ( y, b ) U ( − x, a ) ∗ U ( − y, b ) ∗ SU ( − y, b ) U ( − x, a ) dy dbb dx daa = (cid:90) Aff f ( x, a ) U ( − x, a ) ∗ (cid:90) Aff g ( y, b ) U ( − y, b ) ∗ SU ( − y, b ) dy dbb U ( − x, a ) dx daa = f (cid:63) Aff ( g (cid:63) Aff S ) . Changing the order of integration above is allowed by Fubini’s theorem for Bochner integrals[32, Prop. 1.2.7]. Fubini’s theorem is applicable since (cid:90) Aff (cid:90) Aff | f ( x, a ) | · | g (( z, c ) · ( x, a ) − ) | · (cid:107) U ( − z, c ) ∗ SU ( − z, c ) (cid:107) S dx daa dz dcc is bounded from above by (cid:107) S (cid:107) S (cid:90) Aff | f ( x, a ) | dx daa (cid:90) Aff | g ( z, c ) | dz dcc < ∞ . .2 Relationship With Affine Weyl Quantization The goal of this section is to connect the affine Weyl quantization described in Section 2.3with the convolutions defined in Section 3.1. We first establish a preliminary result describinghow right multiplication on the affine group affects the affine Weyl quantization.
Lemma 3.5.
Let A f ∈ S with affine Weyl symbol f ∈ L r (Aff) . For ( x, a ) ∈ Aff , the affineWeyl symbol of U ( − x, a ) ∗ A f U ( − x, a ) is R ( x,a ) − f .Proof. The result follows from (2.13) and the computation (cid:104) U ( − x, a ) ∗ A f U ( − x, a ) ψ, φ (cid:105) L ( R + ) = (cid:104) A f U ( − x, a ) ψ, U ( − x, a ) φ (cid:105) L ( R + ) = (cid:104) f, W U ( − x,a ) φ,U ( − x,a ) ψ Aff (cid:105) L r (Aff) = (cid:104) f, R ( x,a ) W φ,ψ Aff (cid:105) L r (Aff) = (cid:104) R ( x,a ) − f, W φ,ψ Aff (cid:105) L r (Aff) . We are now ready to prove the first result showing the connection between convolutionand affine Weyl quantization.
Proposition 3.6.
Assume that A f ∈ S with affine Weyl symbol f ∈ L r (Aff) , and let g ∈ L r (Aff) . Then the affine Weyl symbol of g (cid:63) Aff A f is g ∗ Aff f , that is, g (cid:63) Aff A f = A g ∗ Aff f . Proof.
The operator g (cid:63) Aff A f is defined as the S -convergent Bochner integral g (cid:63) Aff A f = (cid:90) Aff g ( x, a ) U ( − x, a ) ∗ A f U ( − x, a ) dx daa . By Proposition 2.1, the map W : S → L r (Aff) given by W ( A f ) = f is unitary. Since boundedoperators commute with convergent Bochner integrals, we have using Lemma 3.5 that W ( g (cid:63) Aff A f ) = (cid:90) Aff g ( x, a ) W ( U ( − x, a ) ∗ A f U ( − x, a )) dx daa = (cid:90) Aff g ( x, a ) R ( x,a ) − W ( A f ) dx daa = g ∗ Aff f. We can also express the convolution of two operators in terms of their affine Weyl symbols.
Proposition 3.7.
Let A f , A g ∈ S with affine Weyl symbols f, g ∈ L r (Aff) . If additionally g ∈ L l (Aff) , then we have A f (cid:63) Aff A g = f ∗ Aff ˇ g, where ˇ g ( x, a ) = g (( x, a ) − ) for ( x, a ) ∈ Aff . roof. Using Proposition 2.1 and Lemma 3.5 we compute that( A f (cid:63) Aff A g )( x, a ) = tr( A f U ( − x, a ) ∗ A g U ( − x, a ))= (cid:104) A f , U ( − x, a ) ∗ A ∗ g U ( − x, a ) (cid:105) S = (cid:104) f, R ( x,a ) − g (cid:105) L r (Aff) = (cid:90) Aff f ( y, b ) g (( y, b ) · ( x, a ) − ) dy dbb = (cid:90) Aff f ( y, b )ˇ g (( x, a ) · ( y, b ) − ) dy dbb = f ∗ Aff ˇ g ( x, a ) . The result follows as ˇ g ∈ L r (Aff) if and only if g ∈ L l (Aff). Of particular interest is the affine Weyl quantization of the coordinate functions f x ( x, a ) := x and f a ( x, a ) := a for ( x, a ) ∈ Aff. Due to the fact that the coordinate functions are not in L r (Aff), we first need to interpret the quantizations A f x and A f a in a rigorous manner. Webegin this task by defining rapidly decaying smooth function and tempered distributions onthe affine group. Definition 3.8.
Let S (Aff) denote the smooth functions f : Aff → C such that( x, ω ) (cid:55)−→ f ( x, e ω ) ∈ S ( R ) . We refer to S (Aff) as the space of rapidly decaying smooth functions (or Schwartz functions )on the affine group.There is a natural topology on S (Aff) induced by the semi-norms (cid:107) f (cid:107) α,β := sup x, ω ∈ R | x | α | ω | α (cid:12)(cid:12)(cid:12) ∂ β x ∂ β ω f ( x, e ω ) (cid:12)(cid:12)(cid:12) , (3.4)for α = ( α , α ) and β = ( β , β ) in N × N . With these semi-norms, the space S (Aff)becomes a Fr´echet space. The space of bounded, anti-linear functionals on S (Aff) is denotedby S (cid:48) (Aff) and called the space of tempered distributions on Aff. Lemma 3.9.
For any f ∈ S (cid:48) (Aff) we can define A f as the map A f : S ( R + ) → S (cid:48) ( R + ) defined by the relation (cid:104) A f ψ, φ (cid:105) S (cid:48) , S = (cid:68) f, W φ,ψ Aff (cid:69) S (cid:48) , S , ψ, φ ∈ S ( R + ) . Additionally, the map f (cid:55)→ A f is injective.Proof. It was shown in [5, Cor. 6.6] that for any φ, ψ ∈ S ( R + ) then W φ,ψ Aff ∈ S (Aff). Hencethe pairing (cid:68) f, W φ,ψ Aff (cid:69) S (cid:48) , S is well defined. 15or the injectivity it suffices to show that A f = 0 implies that f = 0. Let us firstreformulate this slightly: If A f = 0, then we have that (cid:104) A f ψ, φ (cid:105) S (cid:48) , S = (cid:68) f, W φ,ψ Aff (cid:69) S (cid:48) , S = 0for all ψ, φ ∈ S ( R + ). We could conclude that f = 0 if we knew that any g ∈ S (Aff) could beapproximated (in the Fr´echet topology) by linear combinations of elements on the form W φ,ψ Aff for ψ, φ ∈ S ( R + ). To see that this is the case, we translate the problem to the Heisenbergsetting.The Mellin transform M is given by M ( φ )( x ) = M r ( φ )( x ) := (cid:90) ∞ φ ( r ) r − πix drr . Define the functions Ψ and Φ to be Ψ( x ) := ψ ( e x ) and Φ( x ) := φ ( e x ) for ψ, φ ∈ L ( R + ). Areformulation of [5, Lem. 6.4] shows that we have the relation W ψ,φ Aff ( x, a ) = M − y ⊗ M b (cid:32) √ b log( b ) b − (cid:33) πiy F σ W (Ψ , Φ) (cid:0) log( b ) , y (cid:1) ( x, a ) , where W is the cross-Wigner distribution. The correspondence preserves Schwartz functions,due to the term (cid:32) √ b log( b ) b − (cid:33) πiy being smooth with polynomially bounded derivatives. This gives a bijective correspondencebetween W ψ,φ Aff ∈ S (Aff) and W (Ψ , Φ) ∈ S ( R ). As such, the injectivity question is reducedto asking whether the linear span of elements on the form W ( f, g ) for f, g ∈ S ( R ) is densein S ( R ). One way to verify this well-known fact is to note that the map f ⊗ g (cid:55)→ W ( f, g ),where f ⊗ g ( x, y ) = f ( x ) g ( y ), extends to a topological isomorphism on S ( R ), see for instance[23, (14.21)] for the formula of this isomorphism. The density of elements on the form W ( f, g )for f, g ∈ S ( R ) therefore follows as the functions h m ⊗ h n , where { h n } ∞ n =0 are the Hermitefunctions, span a dense subspace of S ( R ) by [43, Thm. V.13]. Example 3.10.
Consider the constant function on the affine group given by 1( x, a ) = 1 forall ( x, a ) ∈ Aff. Then the quantization A is the identity operator since for ψ, φ ∈ S ( R + ) (cid:104) A ψ, φ (cid:105) S (cid:48) , S = (cid:104) , W φ,ψ Aff (cid:105) S (cid:48) , S = (cid:90) Aff W φ,ψ Aff ( x, a ) da dxa = (cid:90) ∞ ψ ( a ) φ ( a ) daa = (cid:104) ψ, φ (cid:105) L ( R + ) . Notice that we used a straightforward generalization of the marginal property of the affineWigner distribution given in (2.15), see the proof of [5, Prop. 3.4] for details.16o motivate the next result, consider the coordinate functions σ x ( x, ω ) := x and σ ω ( x, ω ) := ω for ( x, ω ) ∈ R n . The Weyl quantizations L σ x and L σ ω are the well-known position operator and momentum operator in quantum mechanics. In particular, the commutator[ L σ x , L σ ω ] := L σ x ◦ L σ ω − L σ ω ◦ L σ x is a constant times the identity by [26, Prop. 3.8]. This is precisely the relation for the Liealgebra of the Heisenberg group. In light of this, the following proposition shows that theaffine Weyl quantization has the expected expression for the coordinate functions. Theorem 3.11.
Let f x and f a be the coordinate functions on the affine group. The affineWeyl quantizations A f x and A f a are well-defined as maps from S ( R + ) to S (cid:48) ( R + ) and areexplicitly given by A f x ψ ( r ) = 12 πi rψ (cid:48) ( r ) , A f a ψ ( r ) = rψ ( r ) , ψ ∈ S ( R + ) . In particular, we have the commutation relation [ A f x , A f a ] = 12 πi A f a . This is, up to re-normalization, precisely the Lie algebra structure of aff given in (2.2) .Proof.
Let us begin by computing A f x . We can change the order of integrating by Fubini’stheorem and obtain for ψ, φ ∈ S ( R + ) that (cid:104) A f x ψ, φ (cid:105) S (cid:48) , S = (cid:68) f x , W φ,ψ Aff (cid:69) S (cid:48) , S = (cid:90) ∞−∞ (cid:90) ∞ x (cid:90) ∞−∞ φ ( aλ ( u )) ψ ( aλ ( − u )) e − πixu du da dxa = (cid:90) ∞−∞ (cid:90) ∞ (cid:18)(cid:90) ∞−∞ xe πixu dx (cid:19) ψ ( aλ ( u )) φ ( aλ ( − u )) da dua . Notice that the inner integral is equal to (cid:90) ∞−∞ xe πixu dx = 12 πi δ (cid:48) ( u ) , where (cid:90) ∞−∞ δ (cid:48) ( u ) ψ ( u ) du = ψ (cid:48) (0) . Hence we have the relation (cid:104) A f x ψ, φ (cid:105) S (cid:48) , S = 12 πi (cid:90) ∞ ∂∂u (cid:16) ψ ( aλ ( u )) φ ( aλ ( − u )) (cid:17) (cid:12)(cid:12)(cid:12) u =0 daa . By using the formulas λ (0) = 1 and λ (cid:48) (0) = 1 / (cid:104) A f x ψ, φ (cid:105) S (cid:48) , S = 14 πi (cid:90) ∞ a · (cid:16) ψ (cid:48) ( a ) φ ( a ) − ψ ( a ) φ (cid:48) ( a ) (cid:17) daa . (cid:104) A f x ψ, φ (cid:105) S (cid:48) , S = (cid:90) ∞ (cid:20) πi aψ (cid:48) ( a ) (cid:21) φ ( a ) daa . For A f a we have by similar calculations as above that (cid:104) A f a ψ, φ (cid:105) S (cid:48) , S = (cid:90) ∞−∞ (cid:90) ∞ (cid:18)(cid:90) ∞−∞ · e πixu dx (cid:19) a · ψ ( aλ ( u )) φ ( aλ ( − u )) da dua = (cid:90) ∞−∞ (cid:90) ∞ δ ( u ) (cid:16) a · ψ ( aλ ( u )) φ ( aλ ( − u )) (cid:17) da dua = (cid:90) ∞ aψ ( a ) φ ( a ) daa . The commutation relation follows from straightforward computation.
In this section we introduce the affine Grossmann-Royer operator with the aim of obtainingan affine parity operator analogous to the (Heisenberg) parity operator P in Section 2.2.3.The main reason for this is to obtain affine version of the formulas (2.6) and (2.7) so that wecan describe the affine Weyl quantization through convolution. Recall that the (Heisenberg)Grossmann-Royer operator R ( x, ω ) for ( x, ω ) ∈ R n is defined by the relation W ( f, g )( x, ω ) = (cid:104) R ( x, ω ) f, g (cid:105) L ( R n ) , f, g ∈ L ( R n ) . Analogously, we have the following definition.
Definition 3.12.
We define the affine Grossmann-Royer operator R Aff ( x, a ) for ( x, a ) ∈ Affby the relation W ψ,φ Aff ( x, a ) = (cid:104) R Aff ( x, a ) ψ, φ (cid:105) S (cid:48) , S , ψ, φ ∈ S ( R + ) . We restrict our attention to Schwartz functions for convenience since then W ψ,φ Aff ∈ S (Aff)by [5, Cor. 6.6], and hence have well-defined point values. The Grossmann-Royer operator R Aff ( x, a ) is precisely the affine Weyl quantization of the point mass δ Aff ( x, a ) ∈ S (cid:48) (Aff) for( x, a ) ∈ Aff defined by (cid:104) δ Aff ( x, a ) , f (cid:105) S (cid:48) , S := f ( x, a ) , f ∈ S (Aff) . Since this is also true for the Stratonovich-Weyl operator Ω( x, a ) given in (2.11), it followsthat R Aff ( x, a ) = Ω( x, a ) for all ( x, a ) ∈ Aff. From [18, p. 12] it follows that we have the affine covariance relation U ( − x, a ) ∗ R Aff (0 , U ( − x, a ) = R Aff ( x, a ) . (3.5)The following result, which is a straightforward computation, shows that R Aff ( x, a ) is anunbounded and densely defined operator on L ( R + ).18 emma 3.13. Fix ψ ∈ S ( R + ) and ( x, a ) ∈ Aff . The affine Grossmann-Royer operator R Aff ( x, a ) has the explicit form R Aff ( x, a ) ψ ( r ) = e πixλ − ( ra ) λ − (cid:0) ra (cid:1) (cid:16) − e λ − ( ra ) (cid:17) λ − (cid:0) ra (cid:1) − e λ − ( ra ) · ψ (cid:16) re − λ − ( ra ) (cid:17) , where λ is the function given in (2.3) . We will be particularly interested in the affine parity operator P Aff given by the affineGrossmann-Royer operator at the identity element, that is, P Aff ( ψ )( r ) := R Aff (0 , ψ ( r ) = λ − ( r )(1 − e λ − ( r ) )1 + λ − ( r ) − e λ − ( r ) ψ (cid:16) re − λ − ( r ) (cid:17) , for ψ ∈ S ( R + ). The affine parity operator P Aff is symmetric as an unbounded operator on L ( R + ). Moreover, we see from the relation e λ − ( r ) − λ − ( r ) e λ − ( r ) r that we have the alternative formula P Aff ( ψ )( r ) = λ − ( r )1 − re − λ − ( r ) ψ (cid:16) re − λ − ( r ) (cid:17) . (3.6)An important commutation relation for the (Heisenberg) Grossman-Royer operator R ( x, ω )for ( x, ω ) ∈ R n is given by P ◦ R ( x, ω ) = R ( − x, − ω ) ◦ P. (3.7)The following proposition shows that the analogue of (3.7) breaks down in the affine settingdue to Aff being non-unimodular. As the proof is a straightforward computation, we leavethe details to the reader. Proposition 3.14.
The commutation relation P Aff ◦ R Aff ( x, a ) = R Aff (cid:0) ( x, a ) − (cid:1) ◦ P Aff holds precisely for those ( x, a ) ∈ Aff such that ∆( x, a ) = a = 1 . We will now show that both the function λ in (2.3) and the affine parity operator P Aff are related to the Lambert W function. Recall that the (real) Lambert W function is themultivalued function defined to be the inverse relation of the function f ( x ) = xe x for x ∈ R .The function f ( x ) for x < y ∈ ( − /e,
0) precisely twovalues x , x ∈ ( −∞ ,
0) such that x e x = x e x = y. As the solutions appear in pairs, we can define σ to be the function that permutes thesesolutions, that is, σ ( x ) = x and σ ( x ) = x . For y = − /e there is only one solution to19he equation xe x = y , namely x = −
1. Hence we define σ ( −
1) = −
1. We can represent thefunction σ as σ ( x ) = W ( xe x ) , x < − − , x = − W − ( xe x ) , − < x < , where W , W − are the two branches of the Lambert W function satisfying W ( xe x ) = x, for x ≥ − W − ( xe x ) = x, for x ≤ − . Lemma 3.15.
The inverse of λ is given by λ − ( r ) = log (cid:18) − rσ ( − r ) (cid:19) = σ ( − r ) + r, r > . Proof.
To find the inverse of λ we solve the equation r = λ ( u ) = ue u e u − − ue − u − . A simple computation shows that − r = − u − re − u . Making the substitution v = e − u togetherwith straightforward manipulations shows that − re − r = − rve − rv . (3.8)The trivial solution to (3.8) is given by solving the equation − r = − rv . Checking withthe original equation, this can not give the inverse of λ . We get the first equality from thedefinition of σ together with recalling that u = − log( v ). The final equality follows fromlog (cid:18) − rσ ( − r ) (cid:19) = log (cid:32) − rσ ( − r ) σ ( − r ) e σ ( − r ) − re − r (cid:33) = σ ( − r ) + r. Remark.
A minor variation of the function σ appeared in [18, Section 3] where it was defined bythe relation in Lemma 3.15. The advantage of understanding the connection to the Lambert W function is that properties such as σ ( σ ( x )) = x for every x < Corollary 3.16.
The affine parity operator P Aff can be written as P Aff ( ψ )( r ) = σ ( − r ) + rσ ( − r ) + 1 ψ ( − σ ( − r )) , ψ ∈ S ( R + ) . In particular, we have P Aff ( ψ )(1) = 2 ψ (1) .Proof. The formula for P Aff ( ψ ) is obtained from Lemma 3.15 together with (3.6). To find thevalue P Aff ( ψ )(1), we use (3.6) and the fact that ψ (cid:16) re − λ − ( r ) (cid:17) (cid:12)(cid:12)(cid:12) r =1 = ψ (1) . Hence the claim follows from L’Hopital’s rule sincelim r → λ − ( r ) λ − ( r ) + 1 − r = ( λ − ) (cid:48) (1)( λ − ) (cid:48) (1) − . .5 Operator Convolution for Tempered Distributions This section is all about expressing the affine Weyl quantization of a function f ∈ S (Aff) byusing affine convolution. To be able to do this, we will first define what it means for A f tobe a Schwartz operator. Definition 3.17.
We say that a Hilbert-Schmidt operator A : L ( R + ) → L ( R + ) is a Schwartz operator if the integral kernel A K of A satisfies A K ∈ S ( R + × R + ), that is, if( x, ω ) (cid:55)−→ A K ( e x , e ω ) ∈ S ( R ) . Proposition 3.18.
A Hilbert-Schmidt operator A ∈ S is a Schwartz operator if and only if A = A f for some f ∈ S (Aff) .Proof. Assume that A is a Schwartz operator. In [18, Equation (4.8)] it is shown that theintegral kernel A K of A is related to the affine Weyl symbol f A of A by the formula A K ( r, s ) = (cid:90) ∞−∞ f A (cid:18) x, r − s log( r/s ) (cid:19) e πix log( r/s ) dx. Since the inverse-Fourier transform preserves Schwartz functions, together with the definitionof S ( R + × R + ), we have that( r, s ) (cid:55)−→ f A (cid:18) log( r/s ) , r − s log( r/s ) (cid:19) ∈ S ( R + × R + ) . By performing the change of variable x = log( r/s ) and s = e ω for ω ∈ R we obtain( x, ω ) (cid:55)−→ f A (cid:18) x, e ω e x − x (cid:19) ∈ S ( R ) . Finally, by letting u = log(( e x − /x ) + ω we see that( x, u ) (cid:55)−→ f A ( x, e u ) ∈ S ( R ) , due to the fact that x (cid:55)→ log(( e x − /x ) has polynomial growth.Conversely, assume that A = A f for f ∈ S (Aff). The integral kernel A K is then given by A K ( r, s ) = F − ( f ) (cid:18) log( r/s ) , r − s log( r/s ) (cid:19) . By using that the inverse-Fourier transform F − in the first component preserves S (Aff)together with similar substitutions as previously, we have that A K ∈ S ( R + × R + ).We will use the notation S ( L ( R + )) for all Schwartz operators on L ( R + ). There is anatural topology on S ( L ( R + )) induced by the semi-norms (cid:107) A f (cid:107) α,β := (cid:107) f (cid:107) α,β where (cid:107) · (cid:107) α,β are the semi-norms on S (Aff) given in (3.4). Proposition 3.19.
The affine convolution gives a well-defined map S (Aff) (cid:63) Aff S ( L ( R + )) → S ( L ( R + )) . Moreover, for fixed A ∈ S ( L ( R + )) the map S (Aff) (cid:51) f (cid:55)−→ f (cid:63) Aff A ∈ S ( L ( R + )) is continuous. roof. Let f ∈ S (Aff) and A ∈ S ( L ( R + )). Then A = A g for some g ∈ S (Aff) and wehave by Proposition 3.6 that f (cid:63) Aff A = f (cid:63) Aff A g = A f ∗ Aff g . (3.9)Hence the first statement reduces to showing that the usual affine group convolution is awell-defined map S (Aff) ∗ Aff S (Aff) → S (Aff) . After a change of variables, the question becomes whether the map( x, u ) (cid:55)−→ ( f ∗ Aff g )( x, e u ) = (cid:90) R f ( y, e z ) g ( x − ye u − z , e u − z ) dy dz (3.10)is an element in S ( R ). It is straightforward to check that (3.10) is a smooth function.Moreover, since f and g are both in S (Aff), it suffices to show that (3.10) decays faster thanany polynomial towards infinity; we can then iterate the argument to obtain the requireddecay statements for the derivatives.We claim that sup x,u | x | k | u | l | g ( x − ye u − z , e u − z ) | ≤ A gk,l (1 + | y | ) k (1 + | z | ) l , (3.11)where A gk,l is a constant that depends only on the indices k, l ∈ N and g ∈ S (Aff). To showthis, we need to individually consider three cases: • Assume that we only take the supremum over x and u satisfying 2 | z | ≥ | u | and 2 | y | ≥ | x | .Then clearly (3.11) is satisfied with A gk,l = 2 k + l max | g | . • Assume that we only take the supremum over u satisfying 2 | z | ≤ | u | and let x ∈ R be arbitrary. Then e u − z is outside the interval [ e −| u | / , e | u | / ]. Since g ∈ S (Aff) theleft-hand side of (3.11) will eventually decrease when increasing u . When y ≤ x . When y > x would necessitate an increase of u on the scale of u ∼ ln( x ) tocompensate so that the first coordinate in g does not blow up. However, this again forcesthe second coordinate to grow on the scale of x and we would again, due to g ∈ S (Aff),have that the left hand-side of (3.11) would eventually decrease. • Finally, we can consider taking the supremum over x and u satisfying 2 | z | ≥ | u | and2 | y | ≤ | x | . As this case uses similar arguments as above, we leave the straightforwardverification to the reader.Using (3.11) we have thatsup x,u | x k u l ( f ∗ Aff g )( x, e u ) | ≤ A gk,l (cid:90) R | f ( y, e z ) | (1 + | y | ) k (1 + | z | ) l dy dz < ∞ , (3.12)where the last inequality follows from that f ∈ S (Aff). Finally, the continuity of the map f (cid:55)→ f (cid:63) Aff A follows from (3.9) and (3.12). 22 emark. Notice that the proof of Proposition 3.19 shows that affine convolution between f, g ∈ S (Aff) satisfies f ∗ Aff g ∈ S (Aff). This fact, together with Proposition 3.18, strengthens theclaim that S (Aff) is the correct definition for Schwartz functions on the group Aff.The main result in this section is Theorem 3.21 presented below. To state the resultrigorously, we first need to make sense of the convolution between Schwartz functions g ∈ S (Aff) and the affine parity operator P Aff . As motivation for our definition we will usethe following computation: Let S, T ∈ S with affine Weyl symbols f S , f T ∈ L r (Aff). Fix g ∈ S (Aff) and consider the affine Weyl symbol f g(cid:63) Aff S corresponding to the convolution g (cid:63) Aff S . Then (cid:104) f g(cid:63) Aff S , f T (cid:105) L r (Aff) = (cid:104) g (cid:63) Aff S, T (cid:105) S = (cid:28) S, (cid:90) Aff g ( x, a ) U ( − x, a ) T U ( − x, a ) ∗ dx daa (cid:29) S = (cid:28) f S , (cid:90) Aff g ( x, a ) R ( x,a ) f T dx daa (cid:29) L r (Aff) . With this motivation in mind we get the following definition.
Definition 3.20.
Let S : S ( R + ) → S (cid:48) ( R + ) be the operator with affine Weyl symbol f S ∈ S (cid:48) (Aff) and let g ∈ S (Aff). Then g (cid:63) Aff S is defined by its Weyl symbol f g(cid:63) Aff S ∈ S (cid:48) (Aff)satisfying (cid:104) f g(cid:63) Aff S , h (cid:105) S (cid:48) , S := (cid:28) f S , (cid:90) Aff g ( x, a ) R ( x,a ) h dx daa (cid:29) S (cid:48) , S , for all h ∈ S (Aff).Recall that the injectivity in Lemma 3.9 ensures that the operator S in Definition 3.20is well-defined. The argument to show f g(cid:63) Aff S ∈ S (cid:48) (Aff) is similar to the one presented inProposition 3.19. Hence g (cid:63) Aff S is well-defined. Remark.
We could similarly have defined
S (cid:63) Aff A f for S ∈ S ( L ( R + )) and f ∈ S (cid:48) (Aff) byusing Proposition 3.7. For brevity, we restrict ourselves in the next theorem to the case where S = φ ⊗ ψ for ψ, φ ∈ S (Aff). In this case, we can extend Lemma 3.3 and define( φ ⊗ ψ ) (cid:63) Aff A f := (cid:104) A f U ( − x, a ) ψ, U ( − x, a ) φ (cid:105) S (cid:48) , S . We can now finally state the main theorem in this section.
Theorem 3.21.
The affine Weyl quantization A g of g ∈ S (Aff) can be written as A g = g (cid:63) Aff P Aff , where P Aff is the affine parity operator. Moreover, for ψ, φ ∈ S ( R + ) we have that the affineWeyl symbol W ψ,φ Aff of the rank-one operator ψ ⊗ φ can be written as W ψ,φ Aff = ( ψ ⊗ φ ) (cid:63) Aff P Aff . roof. Recall that the affine parity operator P Aff is the affine Weyl quantization of the pointmeasure δ (0 , ∈ S (cid:48) (Aff). As such, the convolution g (cid:63) Aff P Aff is well-defined with the inter-pretation given in Definition 3.20. The affine Weyl symbol f g(cid:63) Aff P Aff of g (cid:63) Aff P Aff is acting on h ∈ S (Aff) by (cid:104) f g(cid:63) Aff P Aff , h (cid:105) S (cid:48) , S := (cid:28) δ (0 , , (cid:90) Aff g ( x, a ) R ( x,a ) h dx daa (cid:29) S (cid:48) , S = (cid:90) Aff g ( x, a ) h ((0 , · ( x, a )) dx daa = (cid:90) Aff g ( x, a ) h ( x, a ) dx daa = (cid:104) g, h (cid:105) L r (Aff) . Since S (Aff) ⊂ L r (Aff) is dense, we can conclude that f g(cid:63) Aff P Aff = g and thus A g = g (cid:63) Aff P Aff .For the second statement, we get that(( ψ ⊗ φ ) (cid:63) Aff P Aff ) ( x, a ) = (cid:104) P Aff U ( − x, a ) ψ, U ( − x, a ) φ (cid:105) S (cid:48) , S = (cid:104) R Aff ( x, a ) ψ, φ (cid:105) S (cid:48) , S = W ψ,φ Aff ( x, a ) . For operator convolutions on the Heisenberg group, we have from (2.5) the important integralrelation (cid:90) R n S (cid:63) T ( z ) dz = tr( S ) tr( T ) . A similar formula for the integral of operator convolutions will not hold generally in the affinesetting. We therefore search for a class of operators where such a relation does hold: the admissible operators . As a first step, we recall the notion of admissible functions . Definition 4.1.
We say that ψ ∈ L ( R + ) is admissible if (cid:90) ∞ | ψ ( r ) | r drr < ∞ . This definition of admissibility is motivated by the theorem of Duflo and Moore [13], seealso [24]. The
Duflo-Moore operator D − in our setting is formally given by D − ψ ( r ) := ψ ( r ) √ r . It is clear that the Duflo-Moore operator D − is a densely defined, self-adjoint positive operatoron L ( R + ) with a densely defined inverse, namely D ψ ( r ) := √ rψ ( r ) . ψ ∈ L ( R + ) is admissible if and only if D − ψ ∈ L ( R + ) . We will on severaloccasions use the commutation relations D U ( x, a ) = (cid:114) a U ( x, a ) D , U ( x, a ) ∗ D − = √ a D − U ( x, a ) ∗ , ( x, a ) ∈ Aff . (4.1)The following orthogonality relation is a trivial reformulation of the classic orthogonalityrelations for wavelets, see for instance [25]. Proposition 4.2.
Let φ, ψ, ξ, η ∈ L ( R + ) and assume that ψ and η are admissible. Then (cid:90) Aff (cid:104) φ, U ( − x, a ) ∗ ψ (cid:105) L ( R + ) (cid:104) ξ, U ( − x, a ) ∗ η (cid:105) L ( R + ) dx daa = (cid:104) φ, ξ (cid:105) L ( R + ) (cid:104)D − η, D − ψ (cid:105) L ( R + ) . In particular, we have (cid:90) Aff (cid:104) φ, U ( − x, a ) ∗ ψ (cid:105) L ( R + ) (cid:104) ξ, U ( − x, a ) ∗ ψ (cid:105) L ( R + ) dx daa = (cid:104) φ, ξ (cid:105) L ( R + ) (cid:107)D − ψ (cid:107) L ( R + ) . Remark.
By Proposition 4.2, admissibility of ψ ∈ L ( R + ) is equivalent to the condition (cid:90) Aff |(cid:104) ψ, U ( − x, a ) ∗ ψ (cid:105) L ( R + ) | dx daa < ∞ . Our goal is now to extend the notion of admissibility to bounded operators on L ( R + ),with the aim of obtaining a class of operators where a formula for the integral of operatorconvolutions similar to (2.5) holds. We will often use that any compact operator S on L ( R + )has a singular value decomposition S = N (cid:88) n =1 s n ξ n ⊗ η n , N ∈ N ∪ {∞} , (4.2)where { ξ n } Nn =1 and { η n } Nn =1 are orthonormal sets in L ( R + ). The singular values { s n } Nn =1 with s n > N = ∞ . If S is a trace-class operator we have { s n } Nn =1 ∈ (cid:96) ( N ) with (cid:107) S (cid:107) S = (cid:107) s n (cid:107) (cid:96) . Since the admissible functions in L ( R + ) form a densesubspace, we can always find an orthonormal basis consisting of admissible functions.The next result concerns bounded operators D S D for a trace-class operator S . To beprecise, this means that we assume that S maps dom( D − ) into dom( D ), and that the operator D S D defined on dom( D ) extends to a bounded operator. Theorem 4.3.
Let S ∈ S satisfy that D S D ∈ L ( L ( R + )) . For any T ∈ S we have that T (cid:63) Aff D S D ∈ L r (Aff) with (cid:107) T (cid:63) Aff D S D(cid:107) L r (Aff) ≤ (cid:107) S (cid:107) S (cid:107) T (cid:107) S , and (cid:90) Aff T (cid:63) Aff D S D ( x, a ) dx daa = tr( T ) tr( S ) . (4.3)25 roof. We divide the proof into three steps.
Step 1:
We first assume that T = ψ ⊗ φ for ψ, φ ∈ dom( D ). Recall that S can be written inthe form (4.2). From Lemma 3.3 and (4.1) we find that T (cid:63) Aff D S D ( x, a ) = (cid:104) S D U ( − x, a ) ψ, D U ( − x, a ) φ (cid:105) L ( R + ) = 1 a (cid:104) SU ( − x, a ) D ψ, U ( − x, a ) D φ (cid:105) L ( R + ) = N (cid:88) n =1 s n a (cid:104) U ( − x, a ) D ψ, η n (cid:105) L ( R + ) (cid:104) ξ n , U ( − x, a ) D φ (cid:105) L ( R + ) . Integrating with respect to the right Haar measure and using that ( x, a ) (cid:55)→ ( x, a ) − inter-changes left and right Haar measure, we get (cid:90) Aff |(cid:104) U ( − x, a ) D ψ, η n (cid:105) L ( R + ) (cid:104) ξ n , U ( − x, a ) D φ (cid:105) L ( R + ) | a dx daa = (cid:90) Aff |(cid:104) U ( − x, a ) ∗ D ψ, η n (cid:105) L ( R + ) (cid:104) ξ n , U ( − x, a ) ∗ D φ (cid:105) L ( R + ) | dx daa ≤ (cid:18)(cid:90) Aff |(cid:104) U ( − x, a ) ∗ D ψ, η n (cid:105) L ( R + ) | dx daa (cid:19) / (cid:18)(cid:90) Aff |(cid:104) ξ n , U ( − x, a ) ∗ D φ (cid:105) L ( R + ) | dx daa (cid:19) / = (cid:107) ψ (cid:107) L ( R + ) (cid:107) φ (cid:107) L ( R + ) , where the last line uses Proposition 4.2. It follows that the sum in the expression for T (cid:63) Aff D S D ( x, a ) converges absolutely in L r (Aff) with (cid:107) T (cid:63) Aff D S D(cid:107) L r (Aff) ≤ (cid:32) N (cid:88) n =1 s n (cid:33) (cid:107) ψ (cid:107) L ( R + ) (cid:107) φ (cid:107) L ( R + ) = (cid:107) S (cid:107) S (cid:107) T (cid:107) S . Equation (4.3) follows in a similar way by integrating the sum expressing
T (cid:63) Aff D S D andusing Proposition 4.2. Step 2:
We now assume that T = ψ ⊗ φ for arbitrary ψ, φ ∈ L ( R + ). Pick sequences { ψ n } ∞ n =1 , { φ n } ∞ n =1 in dom( D ) converging to ψ and φ , respectively, and let T n = ψ n ⊗ φ n . It isstraightforward to check that T n converges to T in S . By (3.3) this implies that T n (cid:63) Aff D S D converges uniformly to T (cid:63) Aff D S D . On the other hand, T n (cid:63) Aff D S D is a Cauchy sequence in L r (Aff): for m, n ∈ N we find by Step 1 that (cid:107) T n (cid:63) Aff D S D − T m (cid:63) Aff D S D(cid:107) L r (Aff) ≤ (cid:107) ψ n ⊗ φ n (cid:63) Aff D S D − ψ m ⊗ φ n (cid:63) Aff D S D(cid:107) L r (Aff) + (cid:107) ψ m ⊗ φ n (cid:63) Aff D S D − ψ m ⊗ φ m (cid:63) Aff D S D(cid:107) L r (Aff) = (cid:107) ( ψ n − ψ m ) ⊗ φ n (cid:63) Aff D S D(cid:107) L r (Aff) + (cid:107) ψ m ⊗ ( φ n − φ m ) (cid:63) Aff D S D(cid:107) L r (Aff) ≤ (cid:107) S (cid:107) S (cid:107) ψ n − ψ m (cid:107) L ( R + ) (cid:107) φ n (cid:107) L ( R + ) + (cid:107) S (cid:107) S (cid:107) ψ m (cid:107) L ( R + ) (cid:107) φ m − φ n (cid:107) L ( R + ) which clearly goes to zero as m, n → ∞ . This means that T n (cid:63) Aff D S D converges in L r (Aff),and the limit must be T (cid:63) Aff D S D as we already know that T n (cid:63) Aff D S D converges uniformly26o this function. In particular, this implies (cid:107) T (cid:63) Aff D S D(cid:107) L r (Aff) = lim n →∞ (cid:107) T n (cid:63) Aff D S D(cid:107) L r (Aff) ≤ lim n →∞ (cid:107) ψ n (cid:107) L ( R + ) (cid:107) φ n (cid:107) L ( R + ) (cid:107) S (cid:107) S = (cid:107) ψ (cid:107) L ( R + ) (cid:107) φ (cid:107) L ( R + ) (cid:107) S (cid:107) S . Equation (4.3) also follows by taking the limit of (cid:82) Aff T n (cid:63) Aff D S D ( x, a ) dx daa . Step 3:
We now assume that T ∈ S . Consider the singular value decomposition of T givenby T = M (cid:88) m =1 t m ψ m ⊗ φ m for M ∈ N ∪ {∞} . By (3.3) we have, with uniform convergence of the sum, that T (cid:63) Aff D S D = M (cid:88) m =1 t m ψ m ⊗ φ m (cid:63) Aff D S D . (4.4)Notice that Step 2 implies that the convergence is also in L r (Aff), since M (cid:88) m =1 t m (cid:107) ψ m ⊗ φ m (cid:63) Aff D S D(cid:107) L r (Aff) ≤ M (cid:88) m =1 t m (cid:107) ψ m (cid:107) L ( R + ) (cid:107) φ m (cid:107) L ( R + ) (cid:107) S (cid:107) S = (cid:107) T (cid:107) S (cid:107) S (cid:107) S . In particular,
T (cid:63) Aff D S D ∈ L r (Aff). Finally, (4.3) follows by integrating (4.4) and using thatthe sum converges in L r (Aff) and Step 2.The integral relation (4.3) is somewhat artificial in the sense that it introduces D in theintegrand. We will typically be interested in the integral of T (cid:63) Aff S , not of T (cid:63) Aff D S D . Thismotivates the following definition. Definition 4.4.
Let S be a non-zero bounded operator on L ( R + ) that maps dom( D ) intodom( D − ). We say that S is admissible if the composition D − S D − is bounded on dom( D − )and extends to a trace-class operator D − S D − ∈ S .Assume now that S is admissible, and define R := D − S D − . Clearly R maps dom( D − )into dom( D ) as we assume that S maps dom( D ) into dom( D − ). The following corollary istherefore immediate from Theorem 4.3. We also note that it extends [34, Cor. 1] to non-positive, non-compact operators. Corollary 4.5.
Let S ∈ L ( L ( R + )) be an admissible operator. For any T ∈ S we have that T (cid:63) Aff S ∈ L r (Aff) with (cid:107) T (cid:63) Aff S (cid:107) L r (Aff) ≤ (cid:107)D − S D − (cid:107) S (cid:107) T (cid:107) S , and (cid:90) Aff T (cid:63) Aff S ( x, a ) dx daa = tr( T ) tr( D − S D − ) . xample 4.6. A rank-one operator S = η ⊗ ξ for non-zero η, ξ is an admissible operator if andonly if η, ξ ∈ L ( R + ) are admissible functions. Requiring that S maps dom( D ) into dom( D − )clearly implies that η ∈ dom( D − ), i.e. η is admissible. For D − S D − to be trace-class, themap ψ (cid:55)→ (cid:107)D − S D − ψ (cid:107) L ( R + ) = |(cid:104)D − ψ, ξ (cid:105) L ( R + ) | · (cid:107)D − η (cid:107) L ( R + ) , ψ ∈ dom( D − ) , must at least be bounded for (cid:107) ψ (cid:107) L ( R + ) ≤
1. This is bounded if and only if ψ (cid:55)→ (cid:104)D − ψ, ξ (cid:105) L ( R + ) is bounded, which is precisely the condition that ξ ∈ dom (cid:0)(cid:0) D − (cid:1) ∗ (cid:1) = dom( D − ). Hence ournotion of admissibility for operators naturally extends the classical function admissibility. Inthe case of rank-one operators, it follows from Lemma 3.3 and the computationtr( D − ( η ⊗ ξ ) D − ) = (cid:104)D − η, D − ξ (cid:105) L ( R + ) that Corollary 4.5 reduces to Proposition 4.2.When both S and T are admissible trace-class operators, their convolution T (cid:63) Aff S behaveswell with respect to both the left and right Haar measures. Corollary 4.7.
Let S and T be admissible trace-class operators on L ( R + ) . Then the con-volution T (cid:63) Aff S satisfies T (cid:63) Aff S ∈ L r (Aff) ∩ L l (Aff) and (cid:90) Aff T (cid:63) Aff S ( x, a ) dx daa = tr( T ) tr( D − S D − ) , (cid:90) Aff T (cid:63) Aff S ( x, a ) dx daa = tr( S ) tr( D − T D − ) . Proof.
The first equation and the claim that
T (cid:63) Aff S ∈ L r (Aff) is Corollary 4.5. The secondequation and the claim that T (cid:63) Aff S ∈ L l (Aff) follows since T (cid:63) Aff S ( x, a ) = S (cid:63) Aff T (( x, a ) − ) . We now turn to the case where S is a positive compact operator. We first note thatadmissibility in this case becomes a statement about the eigenvectors and eigenvalues of S . Proposition 4.8.
Let S be a non-zero positive compact operator with spectral decomposition S = N (cid:88) n =1 s n ξ n ⊗ ξ n for N ∈ N ∪ {∞} . Then S is admissible if and only each ξ n is admissible and N (cid:88) n =1 s n (cid:107)D − ξ n (cid:107) L ( R + ) < ∞ . roof. We first assume that S is admissible. By linearity and Lemma 3.3 we get for ξ ∈ L ( R + )with (cid:107) ξ (cid:107) L ( R + ) = 1 that ξ ⊗ ξ (cid:63) Aff S ( x, a ) = N (cid:88) n =1 s n |(cid:104) ξ, U ( − x, a ) ∗ ξ n (cid:105) L ( R + ) | . (4.5)Integrating (4.5) using the monotone convergence theorem and Proposition 4.2, we obtain (cid:90) Aff ξ ⊗ ξ (cid:63) Aff S ( x, a ) dx daa = N (cid:88) n =1 s n (cid:107)D − ξ n (cid:107) L ( R + ) . The claim now follows from Corollary 4.5.For the converse, it is clear by the assumption that the operator N (cid:88) n =1 s n ( D − ξ n ) ⊗ ( D − ξ n ) (4.6)is a trace-class operator. It only remains to show that S maps dom( D ) into dom( D − ) andthat D − S D − is given by (4.6). This is easily shown when N is finite, so we do the proof for N = ∞ .The partial sums for ψ ∈ L ( R + ) are denoted by( Sψ ) M := M (cid:88) n =1 s n (cid:104) ψ, ξ n (cid:105) L ( R + ) ξ n , and converge in the sense that ( Sψ ) M → Sψ as M → ∞ . Furthermore, it is clear that ( Sψ ) M is in the domain of D − for each M as each ξ n is admissible. We also have that D − ( Sψ ) M = M (cid:88) n =1 s n (cid:104) ψ, ξ n (cid:105) L ( R + ) D − ξ n . The sequence of partial sums D − ( Sψ ) M also converges in L ( R + ), since by using H¨older’sinequality and Bessel’s inequality we obtain ∞ (cid:88) n =1 s n |(cid:104) ψ, ξ n (cid:105) L ( R + ) |(cid:107)D − ξ n (cid:107) L ( R + ) ≤ (cid:32) ∞ (cid:88) n =1 |(cid:104) ψ, ξ n (cid:105) L ( R + ) | (cid:33) / (cid:32) ∞ (cid:88) n =1 s n (cid:107)D − ξ n (cid:107) L ( R + ) (cid:33) / (cid:46) (cid:107) ψ (cid:107) L ( R + ) (cid:32) ∞ (cid:88) n =1 s n (cid:107)D − ξ n (cid:107) L ( R + ) (cid:33) / . Since D − is a closed operator, we get that Sψ belongs to the domain of D − and D − Sψ = ∞ (cid:88) n =1 s n (cid:104) ψ, ξ n (cid:105) L ( R + ) D − ξ n . φ ∈ dom( D − ), we have that D − S D − φ = ∞ (cid:88) n =1 s n (cid:104)D − φ, ξ n (cid:105) L ( R + ) D − ξ n = ∞ (cid:88) n =1 s n (cid:104) φ, D − ξ n (cid:105) L ( R + ) D − ξ n , so D − S D − agrees with (4.6) on this dense subspace. In fact, they agree on all of L ( R + )since (cid:107)D − S D − φ (cid:107) L ( R + ) ≤ (cid:107) φ (cid:107) L ( R + ) ∞ (cid:88) n =1 s n (cid:107)D − ξ n (cid:107) L ( R + ) , shows that D − S D − extends to a bounded operator.As a consequence of Proposition 4.8, we obtain a compact reformulation of admissibilityfor positive trace-class operators. Corollary 4.9.
Let T be a non-zero positive trace-class operator on L ( R + ) , and let S be anon-zero positive compact operator. If (cid:90) Aff T (cid:63) Aff S ( x, a ) dx daa < ∞ , then S is admissible with tr( D − S D − ) = 1tr( T ) (cid:90) Aff T (cid:63) Aff S ( x, a ) dx daa . In particular, if S is a non-zero, positive trace-class operator, then S is admissible if and onlyif S (cid:63) Aff S ∈ L r (Aff) .Proof. Let S = N (cid:88) n =1 s n ξ n ⊗ ξ n be the spectral decomposition of S . An argument similar to the one giving in the proof ofProposition 4.8 shows that (cid:90) Aff T (cid:63) Aff S ( x, a ) dx daa = tr( T ) N (cid:88) n =1 s n (cid:107)D − ξ n (cid:107) L ( R + ) . The claims now follow immediately from Proposition 4.8.
Although we derived several basic properties of admissible operators in Section 4.1, we havenot given any way to construct such operators in practice. Our construction is based on thefollowing observation: From Proposition 4.8 we know that if S = ∞ (cid:88) n =1 s n ϕ n ⊗ ϕ n
30s a non-zero positive compact operator with ∞ (cid:88) n =1 s n (cid:107)D − ϕ n (cid:107) L ( R + ) < ∞ , then S is admissible. So if we can find an orthonormal basis { ϕ n } ∞ n =1 of admissible functionssuch that we can control the terms (cid:107)D − ϕ n (cid:107) L ( R + ) , then we can construct admissible operatorsas infinite linear combinations of rank-one operators. It turns out that the Laguerre basisworks extremely well in this regard. Definition 4.10.
For fixed α ∈ R + we define the Laguerre basis (cid:110) L ( α ) n (cid:111) ∞ n =0 for L ( R + ) by L ( α ) n ( r ) := (cid:115) n !Γ( n + α + 1) r α +12 e − r L ( α ) n ( r ) , n ∈ N , r ∈ R + , where Γ denotes the gamma function and L ( α ) n denotes the generalized Laguerre polynomials given by L ( α ) n ( r ) := r − α e r n ! d n dr n (cid:0) e − r r n + α (cid:1) = n (cid:88) k =0 ( − k (cid:18) n + αn − k (cid:19) r k k ! . The classical orthogonality relation (cid:90) ∞ x α e − x L ( α ) n ( x ) L ( α ) m ( x ) dx = Γ( n + α + 1) n ! δ n,m , (4.7)for the generalized Laguerre polynomials ensures that the Laguerre bases are orthonormalbases for L ( R + ) for any fixed α ∈ R + . The following result shows that the Laguerre basis isespecially compatible with the Duflo-Moore operator D − . Proposition 4.11.
For any α ∈ R + and n ∈ N we have (cid:13)(cid:13)(cid:13) D − L ( α ) n (cid:13)(cid:13)(cid:13) L ( R + ) = n !Γ( n + α + 1) (cid:90) ∞ e − r r α − (cid:16) L ( α ) n ( r ) (cid:17) dr = 1 α . (4.8) Proof.
The first equality in (4.8) follows from unwinding the definitions. For the secondequality in (4.8), we will use the well-known identity L ( α ) n ( r ) = n (cid:88) j =0 L ( α − j ( r )together with the orthogonality relation (4.7). This gives (cid:90) ∞ e − r r α − (cid:16) L ( α ) n ( r ) (cid:17) dr = n (cid:88) i,j =0 (cid:90) ∞ e − r r α − L ( α − i ( r ) L ( α − j ( r ) dr = n (cid:88) i =0 Γ( i + α ) i != 1 α Γ( n + α + 1) n ! , where the last equality follows from a straightforward induction argument.31he following consequence from Proposition 4.8 shows that we can explicitly constructadmissible operators by using the Laguerre basis. Corollary 4.12.
Let { s n } ∞ n =0 ∈ (cid:96) ( N ) be a sequence of non-negative numbers and let α ∈ R + .Then S := ∞ (cid:88) n =0 s n L ( α ) n ⊗ L ( α ) n is an admissible operator with tr( D − S D − ) = 1 α ∞ (cid:88) n =0 s n . Remark.
The corollary may be considered a reformulation with slightly different proof of thecalculations in [21, Section 3.3], where a resolution of the identity operator is constructedfrom thermal states that are diagonal in the Laguerre basis. We will return to resolutions ofthe identity operator and the relation to admissibility in Section 6.2.
We will now see how admissibility relates to the convolution of a function with an operator.The following result shows that we can use convolutions to generate new admissible operatorsfrom a given admissible operator.
Proposition 4.13.
Let f ∈ L l (Aff) ∩ L r (Aff) be a non-zero positive function. If S is apositive, admissible trace-class operator on L ( R + ) , then so is f (cid:63) Aff S with tr (cid:0) D − ( f (cid:63) Aff S ) D − (cid:1) = (cid:90) Aff f ( x, a ) dx daa tr( D − S D − ) . Proof.
It is clear from (3.2) that f (cid:63) Aff S is a trace-class operator, and positivity follows fromthe definition of the convolution f (cid:63) Aff S . Let T be a non-zero positive trace-class operatoron L ( R + ). It suffices by Corollary 4.9 to show that (cid:90) Aff T (cid:63) Aff ( f (cid:63) Aff S )( y, b ) dy dbb = tr( T ) (cid:90) Aff f ( x, a ) dx daa tr( D − S D − ) . We have that
T (cid:63) Aff ( f (cid:63) Aff S )( y, b ) = tr (cid:18) T U ( − y, b ) ∗ (cid:90) Aff f ( x, a ) U ( − x, a ) ∗ SU ( − x, a ) dx daa U ( − y, b ) (cid:19) = (cid:90) Aff f ( x, a ) tr( T U (( − x, a ) · ( − y, b )) ∗ SU (( − x, a ) · ( − y, b )) dx daa = (cid:90) Aff f ( x, a ) T (cid:63) Aff S (( x, a ) · ( y, b )) dx daa .
32e may then use Fubini’s theorem, which applies by our assumptions on f and S , to showthat (cid:90) Aff T (cid:63) Aff ( f (cid:63) Aff S )( y, b ) dy dbb = (cid:90) Aff f ( x, a ) (cid:90) Aff T (cid:63) Aff S (( x, a ) · ( y, b )) dy dbb dx daa = (cid:90) Aff f ( x, a ) dx daa ∆( x, a ) (cid:90) Aff T (cid:63) Aff S ( y, b ) dy dbb = (cid:90) Aff f ( x, a ) dx daa tr( T ) tr (cid:0) D − S D − (cid:1) , where we used the admissibility of S and Theorem 4.5 in the last line. Remark.
We can give a simple heuristic argument for Proposition 4.13 by ignoring that D − is unbounded as follows: We have by using (4.1) that D − ( f (cid:63) Aff S ) D − = (cid:90) Aff f ( x, a ) D − U ( − x, a ) ∗ SU ( − x, a ) D − dx daa = (cid:90) Aff f ( x, a ) U ( − x, a ) ∗ D − S D − U ( − x, a ) dx daa . Since D − S D − is a trace-class operator, the integral above is a convergent Bochner integraland we obtain the desired equality. In this section we will delve more into how the non-unimodularity of the affine group affectsthe affine Weyl quantization. As we will see, both the left and right Haar measures take onan active role in this picture.
Proposition 4.14.
Let S be an admissible Hilbert-Schmidt operator on L ( R + ) such that itsaffine Weyl symbol f S satisfies f S ∈ L l (Aff) . Then tr (cid:0) D − S D − (cid:1) = (cid:90) Aff f S ( x, a ) dx daa . Proof.
Let T = ϕ ⊗ ϕ for some non-zero ϕ ∈ S ( R + ). Then the affine Weyl symbol of T is f T = W ϕ Aff ∈ S (Aff). We know by Corollary 4.5 that (cid:90) Aff T (cid:63) Aff S ( x, a ) dx daa = tr( T ) tr (cid:0) D − S D − (cid:1) . On the other hand, Fubini’s theorem together with Proposition 3.7 allows us to calculate that (cid:90) Aff T (cid:63) Aff S ( x, a ) dx daa = (cid:90) Aff f T ∗ Aff ˇ f S ( x, a ) dx daa = (cid:90) Aff f T ( y, b ) (cid:90) Aff f S (( y, b )( x, a ) − ) dx daa dy dbb = (cid:90) Aff f T ( y, b ) dy dbb (cid:90) Aff f S ( x, a ) dx daa . (cid:90) Aff f T ( y, b ) dy dbb = (cid:107) ϕ (cid:107) L ( R + ) = tr( T ) . The claim now follows from combining the calculations we have done.
Remark.
Assuming that T is a trace-class operator we have thattr( T ) = (cid:90) Aff f T ( x, a ) dx daa , which follows from a similar proof to the one in Proposition 4.14. This gives the interestingheuristic interpretation that taking D − T D − of an operator T coincides with multiplying f T by a .The following result shows that the affine Wigner distribution satisfies both left and rightintegrability when more is assumed of the input. This should be compared with the Heisenbergcase where the Heisenberg group H n is unimodular. Theorem 4.15.
Assume that φ, ψ, D φ, D ψ ∈ L ( R + ) . Then the affine Wigner distributionsatisfies W φ,ψ Aff ∈ L r (Aff) ∩ L l (Aff) . Proof.
We already know that W φ,ψ Aff is in L r (Aff) by the orthogonality relations (2.14). Usingthe definition of the affine Wigner distribution and Plancherel’s theorem, we have that (cid:107) W φ,ψ Aff (cid:107) L l (Aff) = (cid:90) Aff (cid:12)(cid:12) φ ( aλ ( x )) | | ψ ( aλ ( − x )) (cid:12)(cid:12) dx daa = (cid:90) ∞ (cid:90) ∞ | φ ( v ) | | ψ ( w ) | v − w log( v/w ) dw dvvw , where we used the change of variables v = aλ ( x ) and w = aλ ( − x ) in the last line. By ourassumptions on φ and ψ , it will suffice to show that for all v, w ∈ R + we have the upperbound v − wvw log( v/w ) ≤ · max (cid:26) , v , w , vw (cid:27) . It will be enough by symmetry to consider Λ = { ( v, w ) ∈ R + × R + : v > w } . We havethe decomposition Λ = C ∪ C ∪ C , where C := (cid:26) ( v, w ) ∈ Λ : w ≤ − σ ( − v/ (cid:27) , C := (cid:26) ( v, w ) ∈ Λ : w ≥ − σ ( − /v ) (cid:27) , C := (cid:26) ( v, w ) ∈ Λ : − σ ( − v/ ≤ w ≤ − σ ( − /v ) (cid:27) , where σ is the function appearing in Lemma 3.15.34igure 1: A drawing marking the beginning and end of the different domains. • The level surface g ( v, w ) = ( v − w ) / log( v/w ) = C for C > w = − Cσ (cid:16) − vC (cid:17) . (4.9)On C we are below the level surface (4.9) with C = 2. Notice that (1 , . ∈ C with g (1 , .
5) = log( √ <
2. The continuity of g forces the inequality g ( v, w ) ≤ v, w ) ∈ C . Hence v − wvw log( v/w ) ≤ vw . • Notice that v − wvw log( v/w ) = v − w log((1 /v ) / (1 /w )) . Hence the case of C follows from the previous the argument for C by considering thelevel surface of g (1 /v, /w ) = 1 . • It is straightforward to verify that v > w < v, w ) ∈ C . Hence we obtainfor any ( v, w ) ∈ C that v − wwv log( v/w ) ≤ vwv log(2) ≤ /w. Remark.
The connection from this result to admissibility is that the assumptions boil downto S = D ψ ⊗ D φ being an admissible operator. Remark.
Let A be a Hilbert-Schmidt operator on L ( R + ) with integral kernel A K . Thenone can gauge from the proof of Theorem 4.15 that the affine Weyl symbol f A satisfies f A ∈ L r (Aff) ∩ L l (Aff) if and only if the integral kernel A K satisfies A K ∈ L (cid:18) R + × R + , s − tst log( s/t ) dt ds (cid:19) ∩ L (cid:18) R + × R + , st dt ds (cid:19) . .5 Extending the Setting Except for Section 3.5, we have so far considered convolutions between rather well-behavedfunctions and operators and obtained norm estimates for the norms of L r (Aff), L ∞ (Aff), S and L ( L ( R + )). We have seen that (cid:107) f (cid:63) Aff S (cid:107) S ≤ (cid:107) f (cid:107) L r (Aff) (cid:107) S (cid:107) S , (cid:107) T (cid:63) Aff S (cid:107) L ∞ (Aff) ≤ (cid:107) T (cid:107) L ( L ( R + )) (cid:107) S (cid:107) S . This generalizes these inequalities to other Schatten classes and L p spaces. Proposition 4.16.
Let ≤ p ≤ ∞ and let q be its conjugate exponent given by p − + q − = 1 .If S ∈ S p , T ∈ S q , and f ∈ L r (Aff) , then the following hold:1. f (cid:63) Aff S ∈ S p with (cid:107) f (cid:63) Aff S (cid:107) S p ≤ (cid:107) f (cid:107) L r (Aff) (cid:107) S (cid:107) S p .2. T (cid:63) Aff S ∈ L ∞ (Aff) with (cid:107) T (cid:63) Aff S (cid:107) L ∞ (Aff) ≤ (cid:107) S (cid:107) S p (cid:107) T (cid:107) S q .Proof. For p < ∞ , we can clearly interpret the definition of f (cid:63) Aff S as a convergent Bochnerintegral in S p . Hence the first inequality follows from [32, Prop. 1.2.2]. For p = ∞ , we avoidthe unpleasantness of Bochner integration in non-separable Banach spaces by interpreting f (cid:63) Aff S weakly by (cid:104) f (cid:63) Aff Sψ, φ (cid:105) L ( R + ) = (cid:90) Aff f ( x, a ) (cid:104) SU ( − x, a ) ψ, U ( − x, a ) φ (cid:105) L ( R + ) dx daa , for ψ, φ ∈ L ( R + ). A standard argument shows that f (cid:63) Aff S is a bounded operator with (cid:107) f (cid:63) Aff S (cid:107) L ( L ( R + )) ≤ (cid:107) f (cid:107) L r (Aff) (cid:107) S (cid:107) L ( L ( R + )) . Inequality 2. follows from the H¨older type inequality [45, Thm. 2.8].We have already seen in Section 4.1 that we can say more about operator convolutionswhen one of the operators is admissible. As the next lemma shows, admissibility is also thecorrect condition to ensure that f (cid:63) Aff S defines a bounded operator for all f ∈ L ∞ (Aff). Lemma 4.17.
Let S ∈ S and f ∈ L ∞ (Aff) . Define the operator f (cid:63) Aff D S D weakly for ψ, φ ∈ Dom( D ) by (cid:104) f (cid:63) Aff D S D ψ, φ (cid:105) L ( R + ) = (cid:90) Aff f ( x, a ) (cid:104) S D U ( − x, a ) ψ, D U ( − x, a ) φ (cid:105) L ( R + ) dx daa . (4.10) Then f (cid:63) Aff D S D uniquely extends to a bounded linear operator on L ( R + ) satisfying (cid:107) f (cid:63) Aff D S D(cid:107) L ( L ( R + )) ≤ (cid:107) f (cid:107) L ∞ (Aff) (cid:107) S (cid:107) S . In particular, if R is an admissible operator, then f (cid:63) Aff R ∈ L ( L ( R + )) with (cid:107) f (cid:63) Aff R (cid:107) L ( L ( R + )) ≤ (cid:107) f (cid:107) L ∞ (Aff) (cid:107)D − R D − (cid:107) S . roof. By using (4.1) we get that (cid:104) f (cid:63) Aff D S D ψ, φ (cid:105) L ( R + ) = (cid:90) Aff f ( x, a ) (cid:104) SU ( − x, a ) D ψ, U ( − x, a ) D φ (cid:105) L ( R + ) dx daa = (cid:90) Aff ˇ f ( x, a ) (cid:104) SU ( − x, a ) ∗ D ψ, U ( − x, a ) ∗ D φ (cid:105) L ( R + ) dx daa = (cid:90) Aff ˇ f ( x, a )( S (cid:63) Aff ( D ψ ⊗ D φ ))( x, a ) dx daa . Clearly D ψ ⊗ D φ is an admissible operator with | tr( D − ( D ψ ⊗ D φ ) D − ) | = |(cid:104) ψ, φ (cid:105)| L ( R + ) ≤ (cid:107) ψ (cid:107) L ( R + ) (cid:107) φ (cid:107) L ( R + ) . By Corollary 4.5 we therefore get (cid:12)(cid:12) (cid:104) f (cid:63) Aff D S D ψ, φ (cid:105) L ( R + ) (cid:12)(cid:12) ≤ (cid:107) f (cid:107) L ∞ (Aff) (cid:107) S (cid:107) S (cid:107) ψ (cid:107) L ( R + ) (cid:107) φ (cid:107) L ( R + ) . The density of dom( D ) implies that f (cid:63) Aff D S D extends to a bounded operator on L ( R + ).Armed with Lemma 4.17 and Corollary 4.5, we prove the following result describing L p and S p properties of convolutions with admissible operators. The proof is essentially anapplication of complex interpolation: we refer to [45, Thm. 2.10] and [8, Thm. 5.1.1] for theinterpolation theory of S p and L pr (Aff). Proposition 4.18.
Let ≤ p ≤ ∞ and let q be its conjugate exponent given by p − + q − = 1 .If R ∈ S p , g ∈ L pr (Aff) , and S is an admissible trace-class operator, then:1. g (cid:63) Aff S ∈ S p with (cid:107) g (cid:63) Aff S (cid:107) S p ≤ (cid:107) S (cid:107) /p S (cid:107)D − S D − (cid:107) /q S (cid:107) g (cid:107) L pr (Aff) .2. R (cid:63) Aff S ∈ L pr (Aff) with (cid:107) R (cid:63) Aff S (cid:107) L pr (Aff) ≤ (cid:107) S (cid:107) /q S (cid:107)D − S D − (cid:107) /p S (cid:107) R (cid:107) S p .Proof. For g ∈ L r (Aff) ∩ L ∞ (Aff), we have for p = ∞ that Lemma 4.17 gives (cid:107) g (cid:63) Aff S (cid:107) L ( L ( R + )) ≤ (cid:107)D − S D − (cid:107) S (cid:107) g (cid:107) L ∞ (Aff) . Since we also have (cid:107) g (cid:63) Aff S (cid:107) S ≤ (cid:107) g (cid:107) L r (Aff) (cid:107) S (cid:107) S , the first result follows by complex interpo-lation. For the second claim, if R ∈ S we know from Corollary 4.5 that (cid:107) R (cid:63) Aff S (cid:107) L r (Aff) ≤ (cid:107)D − S D − (cid:107) S (cid:107) R (cid:107) S . The result follows by complex interpolation since (cid:107)
R (cid:63) Aff S (cid:107) L ∞ (Aff) ≤ (cid:107) S (cid:107) S (cid:107) R (cid:107) L ( L ( R + )) . We will for completeness investigate how various notions of affine Fourier transforms fit intoour framework. As we will see, known results from abstract wavelet analysis give connectionsbetween affine Weyl quantization, affine Fourier transforms, and admissibility for operators.37 .1 Affine Fourier Transforms
Definition 5.1.
For f ∈ L l (Aff) we define the (left) integrated representation U ( f ) to be theoperator on L ( R + ) given by U ( f ) ψ := (cid:90) Aff f ( x, a ) U ( x, a ) ψ dx daa , ψ ∈ L ( R + ) . The inverse affine Fourier-Wigner transform F − W ( f ) of f ∈ L r (Aff) is given by F − W ( f ) := U ( ˇ f ) ◦ D , ˇ f ( x, a ) := f (( x, a ) − ) . The inverse affine Fourier-Wigner transform F − W ( f ) of f ∈ L r (Aff) is explicitly given by F − W ( f ) ψ ( s ) = (cid:90) ∞ √ r F ( f )( r, s/r ) ψ ( r ) drr , where F denotes the Fourier transform in the first coordinate and ψ ∈ L ( R + ). Hence theintegral kernel of F − W ( f ) is given by K f ( s, r ) = √ r ( F f )( r, s/r ) , s, r ∈ R + . (5.1)It is straightforward to verify that we have the estimate (cid:107)F − W ( f ) (cid:107) S ≤ (cid:107) f (cid:107) L r (Aff) , for every f ∈ L r (Aff) ∩ L r (Aff). Hence we can extend F − W to be defined on L r (Aff) and wehave that F − W ( f ) ∈ S for any f ∈ L r (Aff). Proposition 5.2.
The inverse affine Fourier-Wigner transform is a unitary transformation F − W : Q → S , where Q := { f ∈ L r (Aff) | ess supp( F ( f )) ⊂ R + × R + } . Proof.
Any function K ∈ L ( R + × R + ) can be written uniquely on the form K f in (5.1) forsome f ∈ Q . Moreover, we have (cid:107) K f (cid:107) L ( R + × R + ) = (cid:115)(cid:90) ∞ (cid:90) ∞ |F f ( r, s/r ) | dr dss = (cid:107) f (cid:107) L r (Aff) . Since there is a norm-preserving correspondence between integral kernels in L ( R + × R + ) andHilbert-Schmidt operators on L ( R + ) , the claim follows.It is straightforward to check that the inverse affine Fourier-Wigner transform F − W satisfiesfor f, g ∈ Q the properties • F − W ( f ) ∗ = F − W (∆ / f ∗ ) , f ∗ ( x, a ) := f (( x, a ) − ); • F − W ( f ∗ Aff g ) = F − W ( f ) ◦ D − ◦ F − W ( g ) = U ( ˇ f ) ◦ F − W ( g ); • U ( x, a ) ◦ F − W ( f ) = F − W ( R ( x,a ) ( f )); 38 F − W ( f ) ◦ U ( x, a ) = F − W (cid:0) √ aL ( x,a ) − ( f ) (cid:1) . Definition 5.3.
The affine Fourier-Wigner transform F W : S → Q is defined to be theinverse of F − W | Q . Remark. • To avoid overly cluttered notation, we have used the symbol F W for both the classicalFourier-Wigner transform in Section 2.2.3, and the affine Fourier-Wigner transform. Itshould be clear from the context which operator we are referring to. • Recall that the right multiplication R acts on elements in L r (Aff) by R ( y,b ) f ( x, a ) = f (( x, a )( y, b ))for ( x, a ) , ( y, b ) ∈ Aff. For a closed subspace
H ⊂ L r (Aff) invariant under R , we write R | H ∼ = U if there exists a unitary map T : H → L ( R + ) satisfying T ◦ R ( x, a ) f = U ( x, a ) ◦ T f, for all f ∈ H and ( x, a ) ∈ Aff. Define L U (Aff) := span {H ⊂ L r (Aff) : R | H ∼ = U } . From [13, Lem. 3] we deduce that L U (Aff) = Q , as both spaces are the image of the Hilbert-Schmidt operators under the Fourier-Wignertransform. Note that [13] uses left Haar measure, but translating to right Haar measureis an easy exercise using that f (cid:55)→ ˇ f is a unitary equivalence from the left regularrepresentation on L l (Aff) to the right regular representation on L r (Aff). Example 5.4.
Let φ, ψ ∈ L ( R + ) with ψ ∈ dom( D ). If f ( x, a ) = (cid:104) φ, U ( x, a ) ∗ D ψ (cid:105) L ( R + ) , onefinds using Proposition 4.2 that f ∈ L r (Aff) and (cid:104)F − W ( f ) ξ, η (cid:105) L ( R + ) = (cid:104) ( φ ⊗ ψ ) ξ, η (cid:105) L ( R + ) for η ∈ L ( R + ) and ξ ∈ dom( D ) . This implies that F − W ( f ) = φ ⊗ ψ , in other words for( x, a ) ∈ Aff that F W ( φ ⊗ ψ )( x, a ) = (cid:104) φ, U ( x, a ) ∗ D ψ (cid:105) L ( R + ) . For the Heisenberg group, the Fourier-Wigner transform has a very convenient expres-sion for trace-class operators, see (2.8). The corresponding expression on the affine group is F W ( A )( x, a ) = tr( A D U ( x, a )), and the next result shows that it holds as long as the objectsin the formula are well-defined. The result is due to F¨uhr in this generality [17, Thm. 4.15],and builds on an earlier result due to Duflo and Moore [13, Cor. 2]. Proposition 5.5 (F¨uhr, Duflo, and Moore) . Let A ∈ S be such that A D − extends to aHilbert-Schmidt operator. Then F W ( A D − )( x, a ) = tr( AU ( x, a )) . (5.2)39 roof. To see how the result follows from [17, Thm. 4.15], we need some terminology regardingdirect integrals, see [17, Section 3.3]. Recall that the Plancherel theorem [17, Thm. 3.48]supplies a measurable field of Hilbert spaces indexed by the dual group {H π } [ π ] ∈ ˆ G . For theaffine group G = Aff, the Plancherel measure is counting measure supported on the twoirreducible representations π ( x, a ) = U ( x, a ) on L ( R + ) and π ( x, a ) = U ( x, a ) on L ( R − ) := L ( R − , r − dr ). So we can construct an element { A [ π ] } [ π ] ∈ ˆ G of the direct integral (cid:90) ⊕ ˆ G HS ( H π ) d ˆ µ ([ π ])by choosing A [ π ] = A D − and A [ π ] = 0 for [ π ] (cid:54) = [ π ]. Inserting this measurable field oftrace-class operators into [17, Thm. 4.15] then gives the conclusion.For f, g ∈ L ( R ) we denote by SCAL g f the scalogram of f with respect to g given bySCAL g f ( x, a ) := |W g f ( x, a ) | where W g f is the continuous wavelet transform W g f ( x, a ) := 1 √ a (cid:90) R f ( t ) g (cid:18) t − xa (cid:19) dt. The following result, which follows from Lemma 3.3 and Example 5.4, gives a connectionbetween the affine Fourier-Wigner transform, affine convolutions, and the scalogram.
Corollary 5.6.
Let f, g ∈ L ( R ) such that ψ := ˆ f and φ := ˆ g are supported in R + and are in L ( R + ) . If ψ is admissible then |F W ( φ ⊗ D − ψ )( x, a ) | = ( φ ⊗ φ ) (cid:63) Aff ( ψ ⊗ ψ )( − x, a ) = 1 a SCAL g f ( x, a ) . (5.3) Remark.
The condition that ψ is admissible in Corollary 5.6 is only necessary for the firstequality in (5.3). Recall that the affine Wigner distribution W ψ Aff is the affine Weyl symbol ofthe rank-one operator ψ ⊗ ψ . If we use Proposition 3.7 together with Corollary 5.6, then werecover [5, Thm. 5.1].Corollary 5.6 shows that we have the simple relation |F W ( A D − )( x, a ) | = A (cid:63) Aff A ( − x, a ) (5.4)for positive rank-one operators A . By Corollary 4.9, admissibility therefore means that F W ( A D − ) ∈ L r (Aff) in this case. For more general operators, (5.4) will no longer hold.However, we still obtain a result relating admissibility to the Fourier-Wigner transform. Notethat in the first statement in Proposition 5.7 if A ∈ S we interpret F W ( A D − ) := tr( AU ( x, a ))if we do not know that A D − extends to a Hilbert-Schmidt operator. Proposition 5.7.
Let A be a trace-class operator on L ( R + ) . Then the following are equiv-alent:1) F W ( A D − ) ∈ L r (Aff) .2) A D − extends from dom( D − ) to a Hilbert-Schmidt operator on L ( R + ) .3) A ∗ A is admissible. roof. The equivalence of 1) and 2) follows from [17, Thm. 4.15], by applying that theoremto the element { A [ π ] } [ π ] ∈ ˆ G of the direct integral (see proof of Proposition 5.5) (cid:90) ⊕ ˆ G HS ( H π ) d ˆ µ ([ π ])given by choosing A [ π ] = A and A [ π ] = 0 for [ π ] (cid:54) = [ π ].The equivalence of 2) and 3) is clear apart from technicalities resulting from the unbound-edness of D − . If we assume 2), then [44, Thm. 13.2] gives that ( A D − ) ∗ = D − A ∗ , wherethe equality includes equality of domains. As the domain of the left term is all of L ( R + )by assumption, this means that the range of A ∗ is contained in dom( D − ) . In particular, A ∗ A maps dom( D ) into dom( D − ), and as we also have D − A ∗ A D − = ( A D − ) ∗ A D − where A D − is Hilbert-Schmidt, A ∗ A satisfies all requirements for being admissible.Conversely, if A ∗ A is admissible, then we have for ψ ∈ dom( D − ) (cid:107) A D − ψ (cid:107) L ( R + ) = (cid:104)D − A ∗ A D − ψ, ψ (cid:105) L ( R + ) ≤ (cid:107)D − A ∗ A D − (cid:107) L ( L ( R + )) (cid:107) ψ (cid:107) L ( R + ) . So A D − extends to a bounded operator, and as this operator satisfies that( A D − ) ∗ A D − = D − A ∗ A D − is trace-class, A D − is a Hilbert-Schmidt operator. Remark.
Recall that we consider F W a Fourier transform of operators. The inequality (cid:107)F W ( A D − ) (cid:107) L ∞ (Aff) ≤ (cid:107) A (cid:107) S and the equality (cid:107) A (cid:107) S = (cid:107)F W ( A ) (cid:107) L r (Aff) might thereforebe interpreted as the endpoints p = ∞ and p = 2 of a Hausdorff-Young inequality, where theappearance of D − suggests that the definition of the Fourier-Wigner transform must dependon p . In fact, a Hausdorff-Young inequality of this kind—formulated in the other direction,i.e. for maps from functions on Aff to operators—was shown in [14, Thm. 1.41] for 1 ≤ p ≤ affine Fourier-Kirillov transform as the map F KO : Q → L r (Aff)given by( F KO f )( x, a ) = √ a (cid:90) R f (cid:18) vλ ( − u ) , e u (cid:19) e − πi ( xu + av ) du dv (cid:112) λ ( − u ) , ( x, a ) ∈ Aff . More information about the Fourier-Kirillov transform can be found in [33]. The followingresult, which is motivated by (2.9) and is a slight generalization of [3, Section VIII.6], showsthat the affine Weyl quantization is intrinsically linked with the Fourier transforms on theaffine group.
Proposition 5.8.
Let A f be a Hilbert-Schmidt operator on L ( R + ) with affine symbol f ∈ L r (Aff) . Then the following diagram commutes: S Q L r (Aff) F W F KO f (cid:55)−→ A f roof. Recall from (5.1) that the integral kernel of F − W ( g ) for g ∈ Q is given by K g ( s, r ) = √ r ( F g )( r, s/r ) , s, r ∈ R + . Hence by using (2.12) and a change of variables, we see that the affine Weyl symbol of F − W ( g )is given at the point ( x, a ) ∈ Aff by (cid:90) ∞−∞ (cid:112) aλ ( − u ) F ( g )( aλ ( − u ) , e u ) e − πixu du = (cid:90) R (cid:112) aλ ( − u ) g ( v, e u ) e − πi ( xu + avλ ( − u )) du dv = √ a (cid:90) R g (cid:18) vλ ( − u ) , e u (cid:19) e − πi ( xu + av ) du dv (cid:112) λ ( − u )= ( F KO g )( x, a ) . Remark. • In [40] the authors define an alternative quantization scheme on general type 1 groups.Their quantization scheme together with the affine Weyl quantization is used in [40] todefine a quantization scheme on the cotangent bundle T ∗ Aff. • Consider A f for some f ∈ L r (Aff) . Inserting f = F KO F W ( A f ) into Proposition 4.14allows us to obtain a formal expression for tr( D − A f D − ) in terms of F W ( A f ): a formalcalculation gives that for sufficiently nice operators A f we havetr( D − A f D − ) = (cid:90) ∞ [ F F W ( A f )]( a, daa / , (5.5)where F is the Fourier transform in the first coordinate. This is similar to a conditionin [21, Cor. 5.2], where finiteness of (5.5) is used as a necessary condition for 1 (cid:63) Aff A f = I L ( R + ) to hold, where 1( x, a ) = 1 for all ( x, a ) ∈ Aff. We will see in Section 6.2 that thisis closely related to admissibility of A f . Unfortunately, the formal calculation leadingto (5.5) does not give clear conditions on A f for the equality to hold. On the Heisenberg group, the Fourier-Wigner transform behaves in many ways like the Fouriertransform on functions. In particular, for f ∈ L ( R n ) and S, T ∈ S ( R n ) we get the decouplingequations F W ( f (cid:63) S ) = F σ ( f ) F W ( S ) , F σ ( S (cid:63) T ) = F W ( S ) F W ( T ) , (5.6)where F σ denotes the symplectic Fourier transform and F W denotes the classical Fourier-Wigner transform introduced in Section 2.2.3. Although the affine version of (5.6) does nothold, one can develop as a special case of [17, Thm. 4.12] a version of Bochner’s theorem forthe affine Fourier-Wigner transform. This is analogous to the quantum Bochner theorem [46,Prop. 3.2] for the Heisenberg group.Bochner’s classical theorem [16, Thm. 4.19] characterizes functions that are Fourier trans-forms of positive measures. The Bochner theorem for the affine Fourier-Wigner transformanswers the following question: Which functions on Aff are of the form F W ( S ), where S is apositive trace-class operator? As in Bochner’s classical theorem, it turns out that the correctnotion to consider is functions of positive type. Recall that a function f : Aff → C is a42 unction of positive type if for any finite selection of points Ω := { ( x , a ) , . . . , ( x n , a n ) } ⊂ Affthe matrix A Ω with entries ( A Ω ) i,j := f (( x i , a i ) − ( x j , a j ))is positive semi-definite. Before stating the general result we consider an illuminating specialcase. Example 5.9.
Assume that A = φ ⊗ ψ is a rank-one operator where φ, ψ ∈ L ( R + ). We willshow that F W ( A D − )( x, a ) = (cid:104) U ( x, a ) φ, ψ (cid:105) L ( R + ) (5.7)is a function of positive type on Aff if and only if A is a positive operator. If A is positive, thena standard fact [16, Prop. 3.15] shows that (5.7) is a function of positive type. Conversely, wehave from [16, Cor. 3.22] that F W ( φ ⊗ ψ D − )(( x, a ) − ) = F W ( ψ ⊗ φ D − )( x, a ) = F W ( φ ⊗ ψ D − )( x, a ) . Hence (cid:104) U ( x, a ) φ, ψ (cid:105) L ( R + ) = (cid:104) U ( x, a ) ψ, φ (cid:105) L ( R + ) and it follows from [22, Thm. 4.2] that φ = c · ψ for some c ∈ C . We can conclude from [16, Cor. 3.22] that c ≥ F W ( cψ ⊗ ψ D − )(0 ,
1) = c · (cid:107) ψ (cid:107) L ( R + ) ≥ . We are now ready to state the main result regarding positivity. This result is actually,when interpreted correctly, a special case of the general result [17, Thm. 4.12].
Theorem 5.10.
Let A be a trace-class operator on L ( R + ) . Then A is a positive operator ifand only if the function F W ( A D − )( x, a ) = tr( AU ( x, a )) is of positive type on Aff .Proof. We use the same notation as in the proof of Proposition 5.5. For G = Aff, the abstractresult in [17] says that if { A [ π ] } [ π ] ∈ ˆ G ∈ (cid:90) ⊕ ˆ G HS ( H π ) d ˆ µ ([ π ])consists of trace-class operators, then A [ π ] is positive a.e. with respect to ˆ µ if and only if thefunction (cid:82) ˆ G tr( A [ π ] π ( g ) ∗ ) d ˆ µ ([ π ]) is of positive type.As in the proof of Proposition 5.7, we pick A [ π ] = A and A [ π ] = 0 for [ π ] (cid:54) = [ π ]. Theresulting section consists of positive operators for a.e. [ π ] if and only if A is positive. By theabstract result in [17], this happens if and only if (cid:90) ˆ G tr( A [ π ] π ( g ) ∗ ) d ˆ µ ([ π ]) = tr( AU ( x, a ) ∗ )is a function of positive type. The definition of functions of positive type gives that this isequivalent to tr( AU ( x, a )) being of positive type.43 Examples
In this section, we show how the theory developed in this paper provides a common frameworkfor various operators and functions studied by other authors. We also introduce an analogueof the Cohen class of time-frequency distributions for the affine group, and deduce its relationto the previously studied affine quadratic time-frequency representations . There is no general consensus of a localization operator in the affine setting. We will use thefollowing definition based on the convolution framework.
Definition 6.1.
Let f ∈ L r (Aff) and ϕ ∈ L ( R + ). We say that A = f (cid:63) Aff ( ϕ ⊗ ϕ )is an affine localization operator on L ( R + ).Inequality (3.2) shows that an affine localization operator A is a trace-class operator on L ( R + ) with (cid:107) A (cid:107) S ≤ (cid:107) f (cid:107) L r (Aff) (cid:107) ϕ (cid:107) L ( R + ) . Moreover, Proposition 4.13 implies that A is admissible whenever ϕ is admissible and f ∈ L l (Aff) ∩ L r (Aff) . We will now see that the affine localization operators are naturally unitarily equivalent tothe more commonly defined localization operators on the Hardy space H ( R ). Recall thatthe space H ( R ) is the subspace of L ( R ) consisting of elements ψ whose Fourier transform F ψ is supported on R + . Note that the composition DF is a unitary map from H ( R ) to L ( R + ). An admissible wavelet ξ ∈ H ( R ) satisfies by definition that c ξ := (cid:90) ∞ |F ( ξ )( ω ) | ω dω < ∞ . In other words, DF ξ ∈ L ( R + ) is an admissible function in the sense of Definition 4.1. In [47,Thm. 18.13] the localization operator A ξf on H ( R ), given an admissible wavelet ξ ∈ H ( R )and f ∈ L l (Aff), is defined by A ξf ψ = c ξ (cid:90) Aff f ( x, a ) (cid:104) ξ, π ( x, a ) ξ (cid:105) H ( R ) π ( x, a ) ξ dx daa , ξ ∈ H ( R ) , where π acts on H ( R ) by π ( x, a ) ξ ( t ) = 1 √ a ξ (cid:18) t − xa (cid:19) , ψ ∈ H ( R ) . (6.1)The next proposition is straightforward and relates operators on the form A ξf with affinelocalization operators. Proposition 6.2.
Consider f ∈ L l (Aff) and an admissible wavelet ξ ∈ H ( R ) . Then ( DF ) A ξf ( DF ) ∗ = c ξ · ˇ f (cid:63) Aff ( DF ξ ⊗ DF ξ ) . emark.
1. From Proposition 6.2 it follows that Proposition 4.18 is a generalization of the result[47, Thm. 18.13].2. In [12], Daubechies and Paul define localization operators in the same way as in [47],except that they use π ( − x, a ) instead of π ( x, a ) in (6.1) and consider symbols f on thefull affine group Aff F = R × R ∗ . The eigenfunctions and eigenvalues of the resultinglocalization operators acting on L ( R ) are studied in detail in [12] when the window isrelated to the first Laguerre function, and f = χ Ω C whereΩ C := { ( x, a ) ∈ Aff : | ( x, a ) − (0 , C ) | ≤ ( C − } . The corresponding inverse problem, i.e. conditions on the eigenfunctions of the localiza-tion operator that imply that Ω = Ω C , is studied in [1].3. Localization operators with windows related to Laguerre functions have also been ex-tensively studied by Hutn´ık, see for instance [29, 30, 31], with particular emphasis onsymbols f depending only on either x or a . When f ( x, a ) = f ( a ), it is shown thatthe resulting localization operator is unitarily equivalent to multiplication with somefunction γ f . This correspondence allows properties of the localization operator to bededuced from properties of γ f . Operators of the form f (cid:63) Aff S form the basis of the study of covariant integral quantizationsby Gazeau and his collaborators in [2, 6, 7, 19, 20, 21]. Apart from differing conventionsthat we clarify at the end of this section, covariant integral quantizations on Aff are maps Γ S sending functions on Aff to operators given byΓ S ( f ) = f (cid:63) Aff S, for some fixed operator S . By varying S we obtain several quantization maps Γ with propertiesdepending on the properties of S . Examples of such quantization procedures with a differentparametrization of Aff are studied in [21, 7]. Their approach is to define S either by F W ( S )or by its kernel as an integral operator, and deduce conditions on this function that ensuresthe condition 1 (cid:63) Aff S = I L ( R + ) . Example 6.3.
The affine Weyl quantization is an example of a covariant integral quantizationΓ S , where S is not a bounded operator. It corresponds to choosing S = P Aff by Theorem3.21. Remark.
The example above leads to a natural question: could there be other operators P such that f (cid:63) Aff P behaves as an affine analogue of Weyl quantization? Since Weyl quantizationon R n is given by convolving with the parity operator, a natural guess is P ψ ( r ) = ψ (1 /r ) , ψ ∈ L ( R + ) . P ( f ) = f (cid:63) Aff P has been studied by Gazeau and Murenzi in[21, Sec. 7]. It has the advantage that P is a bounded operator, but unfortunately by [21,Prop. 7.5] it does not satisfy the natural dequantization rule f = Γ P ( f ) (cid:63) Aff P. We also mention that Gazeau and Bergeron have shown that this choice of P is merely aspecial case corresponding to ν = − / P ν of operators defining possible affineversions of the Weyl quantization [7, Sec. 4.5].In quantization theory one typically wishes that the domain of Γ S contains L ∞ (Aff). This,by Lemma 4.17, leads us to chose S = D T D for some trace-class operator T . In particular,one requires that Γ S (1) = I L ( R + ) , which can be easily satisfied as the following propositionshows. Proposition 6.4.
Let T be a trace-class operator on L ( R + ) . Then (cid:63) Aff D T D = tr( T ) I L ( R + ) . Proof.
Let ψ, φ ∈ dom( D ). We have by (4.10) that (cid:104) (cid:63) Aff D T D ψ, φ (cid:105) L ( R + ) = (cid:90) Aff (cid:104) U ( − x, a ) ∗ D T D U ( − x, a ) ψ, φ (cid:105) L ( R + ) dx daa = (cid:90) Aff T (cid:63) Aff ( D ψ ⊗ D φ ) dx daa = tr( T ) (cid:104) ψ, φ (cid:105) L ( R + ) , where the last equality uses Theorem 4.3.Following the terminology used by Gazeau et al., we have a resolution of the identityoperator of the form I L ( R + ) = Γ D T D (1) = (cid:90) Aff U ( − x, a ) ∗ D T D U ( − x, a ) dx daa , where tr( T ) = 1 and the integral has the usual weak interpretation.Given a positive trace-class operator T with tr( T ) = 1, we know thatΓ D T D ( f ) = f (cid:63) Aff D T D defines a bounded map Γ D T D : L ∞ (Aff) → L ( L ( R + )) with Γ D T D (1) = I L ( R + ) . Moreover,Γ D T D maps positive functions to positive operators and by a variation of Lemma 3.5 satisfiesthe covariance property U ( − x, a ) ∗ Γ D T D ( f ) U ( − x, a ) = Γ( R ( x,a ) − f ) . The following result, which is a modification of the remark given at the end of [34], shows aremarkable converse to these observations.
Theorem 6.5.
Let
Γ : L ∞ (Aff) → L ( L ( R + )) be a linear map satisfying . Γ sends positive functions to positive operators,2. Γ(1) = I L ( R + ) ,3. Γ is continuous from the weak* topology on L ∞ (Aff) (as the dual space of L r (Aff) ) tothe weak* topology on L ( L ( R + )) ,4. U ( − x, a ) ∗ Γ( f ) U ( − x, a ) = Γ( R ( x,a ) − f ) .Then there exists a unique positive trace-class operator T with tr( T ) = 1 such that Γ( f ) = f (cid:63) Aff D T D . Proof.
The map Γ (cid:55)→ Γ l where Γ l ( f ) = Γ( ˇ f ) is a bijection from maps Γ satisfying the fourassumptions to maps Γ l satisfyingi) Γ l sends positive functions to positive operators,ii) Γ l (1) = I L ( R + ) ,iii) Γ l is continuous from the weak* topology on L ∞ (Aff) (as the dual space of L l (Aff)) tothe weak* topology on L ( L ( R + )),iv) U ( − x, a ) ∗ Γ l ( f ) U ( − x, a ) = Γ l ( L ( x,a ) − f ).The remark in [34] applied to G = Aff and U ( − x, a ) says that if a map Γ l satisfies i)-iv) thenit must be given for ψ, φ ∈ dom( D ) by (cid:104) Γ l ( f ) ψ, φ (cid:105) L ( R + ) = (cid:90) Aff f ( x, a ) (cid:104) U ( − x, a ) T U ( − x, a ) ∗ D ψ, D φ (cid:105) L ( R + ) dx daa , for some trace-class operator T as in the theorem. The relation (4.1) gives that (cid:104) Γ l ( f ) ψ, φ (cid:105) L ( R + ) = (cid:90) Aff f ( x, a ) (cid:104) U ( − x, a ) D T D U ( − x, a ) ∗ ψ, φ (cid:105) L ( R + ) dx daa = (cid:90) Aff ˇ f ( x, a ) (cid:104) U ( − x, a ) ∗ D T D U ( − x, a ) ψ, φ (cid:105) L ( R + ) dx daa . Hence Γ l ( f ) = ˇ f (cid:63) Aff D T D and the result follows. Quantization using admissible trace-class operators
As we have mentioned, the properties of the quantization map Γ( f ) = f (cid:63) Aff S depend onthe properties of S . From Lemma 4.17 we know that if S is admissible, i.e. we can write S = D T D for some trace-class operator T , then Γ S : L ∞ (Aff) → L ( L ( R + )) is bounded. Ifwe further assume that S is a trace-class operator, then Proposition 4.18 shows that Γ S isbounded from L pr (Aff) to S p for all 1 ≤ p ≤ ∞ . In this sense, the ideal class of covariantintegral quantizations Γ S are those given by admissible trace-class operators. Example 6.6. If ϕ ∈ L ( R + ) is an admissible function, then ϕ ⊗ ϕ is an admissible oper-ator. The resulting quantization Γ ϕ ⊗ ϕ is then a special case of the quantization proceduresintroduced by Berezin [4]; Berezin calls f the contravariant symbol of Γ ϕ ⊗ ϕ ( f ). In this sense,the quantization procedures Γ S for admissible S generalize Berezin’s contravariant symbols.47 elation to the Conventions of Gazeau and Murenzi Gazeau and Murenzi [21] work with another parametrization of the affine group, namelyΠ + := R + × R where the group operation between ( q , p ) , ( q , p ) ∈ Π + is given by( q , p ) · ( q , p ) := ( q q , p /q + p ) . There is a unitary representation U G : Π + → U ( L ( R + , dr )) given by U G ( q, p ) ψ ( r ) = (cid:114) q e ipr ψ ( r/q ) = (cid:114) q U ( p/ π, /q ) ψ ( r ) . Given a function ˜ f on Π + and an operator S on L ( R + , dr ), Gazeau and Murenzi define(note that the adjoint is now with respect to L ( R + , dr ), not L ( R + )) A S ˜ f := 1 C S (cid:90) ∞−∞ (cid:90) ∞ ˜ f ( q, p ) U G ( q, p ) SU G ( q, p ) ∗ dq dp, where we assume that S satisfies (cid:90) ∞−∞ (cid:90) ∞ U G ( q, p ) SU G ( q, p ) ∗ dq dp = C S · I L ( R + ,dr ) . The next proposition is straightforward and shows that Gazeau and Murenzi’s framework iseasily related to our affine operator convolutions.
Proposition 6.7.
Let S be an operator on L ( R + , dr ) . Then D − S D is an operator on L ( R + , r − dr ) and D A S ˜ f D − = 2 πC S f (cid:63) Aff ( D S D − ) , where f ( x, a ) = ˜ f ( a, πxa ) for ( x, a ) ∈ Aff . The cross-Wigner distribution W ( ψ, φ ) of ψ, φ ∈ L ( R n ) is known to have certain undesirableproperties. A typical example is that one would like to interpret W ( ψ, φ ) as a probabilitydistribution, but W ( ψ, φ ) is seldom a non-negative function as shown by Hudson in [28]. Toremedy this, Cohen introduced in [11] a new class of time-frequency distributions Q f givenby Q f ( ψ, φ ) := W ( ψ, φ ) ∗ f, (6.2)where f is a tempered distribution on R n . In light of our setup, it is natural to investigatethe affine analogue of the Cohen class. Definition 6.8.
We say that a bilinear map Q : L ( R + ) × L ( R + ) → L ∞ (Aff) belongs to the affine Cohen class if Q = Q S for some S ∈ L ( L ( R + )), where Q S ( ψ, φ )( x, a ) := ( ψ ⊗ φ ) (cid:63) Aff S ( x, a ) = (cid:104) SU ( − x, a ) ψ, U ( − x, a ) φ (cid:105) L ( R + ) . We will write Q S ( ψ ) := Q S ( ψ, ψ ) .
48y Proposition 3.7 we get for S = A f that Q S ( ψ, φ ) = W ψ,φ Aff ∗ Aff ˇ f , (6.3)which shows that our definition of the affine Cohen class is a natural analogue of (6.2).It is straightforward to verify that Q S ( ψ, φ ) is a continuous function on Aff for all ψ, φ ∈ L ( R + ) and S ∈ L ( L ( R + )). Since the affine Cohen class is defined in terms of the operatorconvolutions, we get some simple properties: The statements 1 and 2 in Proposition 6.9follow from Proposition 4.18 and Corollary 4.5. Statement 3 is a simple calculation and thelast statement follows from a short polarization argument. Proposition 6.9.
Let S ∈ L ( L ( R + )) . Then for ψ, φ ∈ L ( R + ) we have the followingproperties:1. The function Q S ( ψ, φ ) satisfies (cid:107) Q S ( ψ, φ ) (cid:107) L ∞ (Aff) ≤ (cid:107) S (cid:107) L ( L ( R + )) (cid:107) ψ (cid:107) L ( R + ) (cid:107) φ (cid:107) L ( R + ) .
2. If S is admissible, then Q S ( ψ, φ ) ∈ L r (Aff) and (cid:90) Aff Q S ( ψ, φ )( x, a ) dx daa = (cid:104) ψ, φ (cid:105) L ( R + ) tr( D − S D − ) .
3. We have the covariance property Q S ( U ( − x, a ) ψ, U ( − x, a ) φ )( y, b ) = Q S ( ψ, φ )(( y, b ) · ( x, a )) (6.4) for all ( x, a ) , ( y, b ) ∈ Aff .4. The function Q S ( ψ, ψ ) is (real-valued) positive for all ψ ∈ L ( R + ) if and only if S is(self-adjoint) positive. Example 6.10.
1. For ψ, φ ∈ L ( R + ) we have Q φ ⊗ φ ( ψ )( x, a ) = |(cid:104) ψ, U ( − x, a ) ∗ φ (cid:105) L ( R + ) | , which by Corollary 5.6 is simply a Fourier transform away from being a scalogram.2. If we relax the requirement that S is bounded in Definition 6.8, then it follows fromTheorem 3.21 that Q P Aff ( ψ ) = W ψ Aff for ψ ∈ S ( R + ) . Hence the affine Wigner distribution can be represented as a (general-ized) affine Cohen class operator. If we define an alternative affine Weyl quantizationusing an operator P as in Section 6.2, then it is clear that Q P gives an alternativeWigner function. See [21, Sec. 7.2] for the case of P ψ ( r ) = ψ (1 /r ).The covariance property (6.4) and some rather weak continuity conditions completelycharacterize the affine Cohen class, as is shown in the following result.49 roposition 6.11. Let Q : L ( R + ) × L ( R + ) → L ∞ (Aff) be a bilinear map. Assume that forall ψ, φ ∈ L ( R + ) we know that Q ( ψ, φ ) is a continuous function on Aff that satisfies (6.4) and the estimate | Q ( ψ, φ )(0 , | (cid:46) (cid:107) ψ (cid:107) L ( R + ) (cid:107) φ (cid:107) L ( R + ) . Then there exists a unique S ∈ L ( L ( R + )) such that Q = Q S .Proof. By assumption, the map ( ψ, φ ) (cid:55)→ Q ( ψ, φ )(0 ,
1) is a bounded bilinear form. Hencethere exists a unique bounded operator S such that (cid:104) Sψ, φ (cid:105) L ( R + ) = Q ( ψ, φ )(0 , . To see that Q = Q S , note that we have Q ( ψ, φ )( x, a ) = Q ( U ( − x, a ) ψ, U ( − x, a ) φ )(0 , (cid:104) SU ( − x, a ) ψ, U ( − x, a ) φ (cid:105) L ( R + ) = Q S ( ψ, φ )( x, a ) . At this point we have seen that operators S define a quantization procedure Γ S ( f ) = f (cid:63) Aff S as in Section 6.2, and an affine Cohen class distribution Q S . The connection betweenthese concepts is provided by the next proposition. Proposition 6.12.
Let S be a positive, compact operator on L ( R + ) and let f ∈ L r (Aff) be a positive function. Then f (cid:63) Aff S is a positive, compact operator. Denote by { λ n } ∞ n =1 itseigenvalues in non-increasing order with associated orthogonal eigenvectors { φ n } ∞ n =1 . Then λ n = max (cid:107) ψ (cid:107) =1 (cid:26)(cid:90) Aff f ( x, a ) Q S ( ψ, ψ )( x, a ) dx daa : ψ ⊥ φ k for k = 1 , . . . , n − (cid:27) . Proof.
The integral defining f (cid:63) Aff S is a Bochner integral of compact operators converging inthe operator norm, hence it defines a compact operator. It is straightforward to check that f (cid:63) Aff S is also a positive operator. Furthermore, for ψ ∈ L ( R + ) we have (cid:104) f (cid:63) Aff Sψ, ψ (cid:105) L ( R + ) = (cid:90) Aff f ( x, a ) (cid:104) SU ( − x, a ) ψ, U ( − x, a ) ψ (cid:105) L ( R + ) dx daa = (cid:90) Aff f ( x, a ) Q S ( ψ, ψ )( x, a ) dx daa . The result therefore follows from Courant’s minimax theorem [35, Thm. 28.4].
Example 6.13.
Let us consider a localization operator χ Ω (cid:63) Aff ϕ ⊗ ϕ for ϕ ∈ L ( R + ) and acompact subset Ω ⊂ Aff. The first eigenfunction φ of this operator maximizes the quantity (cid:104) χ Ω (cid:63) Aff ( ϕ ⊗ ϕ ) φ , φ (cid:105) L ( R + ) = (cid:90) Ω |(cid:104) ϕ , U ( − x, a ) ∗ ϕ (cid:105) L ( R + ) | dx daa . Hence in this sense, the eigenfunctions are the best localized functions in Ω, which explainsthe terminology of localization operators. 50 .3.1 Relation to the Affine Quadratic Time-Frequency Representations
The signal processing literature contains a wealth of two-dimensional representations of sig-nals. Among them we find the affine class of quadratic time-frequency representations , see[41]. A member of the affine class of quadratic time-frequency representations is a map sendingfunctions ψ on R to a function Q A Φ ( ψ ) on R given by Q A Φ ( ψ )( x, a ) = 1 a (cid:90) ∞−∞ (cid:90) ∞−∞ Φ( t/a, s/a ) e πix ( t − s ) ψ ( t ) ψ ( s ) dt ds for some kernel function Φ on R . There are clearly a few differences between our setup andthe affine class of quadratic time-frequency representations. The domain of the affine classconsists of functions on R , whereas the affine Cohen class acts on functions on R + . For afunction ψ on R + we therefore define ψ ( t ) = (cid:40) ψ ( t ) t >
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Proposition 6.14.
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Department of Mathematical Sciences, Norwegian University of Science and Technology,7491 Trondheim, Norway.
E-mail addresses : [email protected] , [email protected] , [email protected] ,and [email protected]@ntnu.no