A Primer on Coorbit Theory -- From Basics to Recent Developments
aa r X i v : . [ m a t h . F A ] J a n A Primer on Coorbit Theory
From Basics to Recent Developments
Eirik Berge
Abstract
Coorbit theory is a powerful machinery that constructs a family of Banach spaces, the so-called coorbit spaces , from well-behaved unitary representations of locally compact groups. Acore feature of coorbit spaces is that they can be discretized in a way that reflects the geometryof the underlying locally compact group. Many established function spaces such as modulationspaces , Besov spaces , Sobolev-Shubin spaces , and shearlet spaces are examples of coorbit spaces.The goal of this survey is to give an overview of coorbit theory with the aim of presenting themain ideas in an accessible manner. Coorbit theory is generally seen as a complicated theory,filled with both technicalities and conceptual difficulties. Faced with this obstacle, we feel obligedto convince the reader of the theory’s elegance. As such, this survey is a showcase of coorbittheory and should be treated as a stepping stone to more complete sources.
Whenever a new mathematical theory is developed, one of two things usually happens: On the onehand, the theory might not be sufficiently interesting. Together with the failure to generate non-trivial results in well-established special cases, this signals a premature end. On the other hand, anewly developed theory might succeed in these endeavours. What follows is a period of flourishing,where researchers from related fields develop the theory to its fullest potential. However, there is athird and more disheartening possibility as well; the theory is wonderful in all regards but is largelyleft unnoticed by the mathematical community. This was the case for the theory of coorbit spaces,developed in the late ’80s in a series of papers [28, 29, 30] by Hans Georg Feichtinger and KarlheinzGr¨ochenig. However, with the turn of the century, interest in coorbit spaces has been growingrapidly. This is due to a plethora of reasons, the most obvious one being the emergence of time-frequency analysis as a central topic in modern harmonic analysis. Many results in time-frequencyanalysis can be either proven or illuminated by the constructions in coorbit theory.The goal of this survey is to provide an introduction to coorbit theory aimed at non-experts.We have tried to strike a balance between providing sufficient details, while at the same time prior-itizing concepts over technicalities. The original papers on coorbit theory are, although insightful,admittedly difficult for novices to digest. More recent sources, e.g. [53, 63, 16], are either not fullydevoted to coorbit theory or include technicalities that distract most beginners from the core ideas.This is not intended as critique of the above sources as their main aim is to derive new results. Infact, we have the privilege of dwelling on pedagogical points precisely because we do not aim fornovelty. We hope this survey can establish a natural starting point to learn coorbit theory for bothstudents and researchers in neighboring fields. 1 verview:
Before embarking, we give a brief overview of what coorbit theory is all about. Thisrequires the usage of terminology that might be unfamiliar to the reader; if this causes bewilderment,then skip this part for now and return to it once you have finished reading Chapter 2. We beginwith a unitary representation π : G → H π of a locally compact group G on a Hilbert space H π .Consider the wavelet transform W g : H π → L ∞ ( G ) , W g ( f )( x ) := h f, π ( x ) g i H π , where g, f ∈ H π and x ∈ G . Under some assumptions on the representation π and the element g ∈ H π , the transformation W g is actually an isometry from H π to the Hilbert space L ( G ). Theinner mechanics of coorbit theory deals with the following two points: • We construct a collection C o p ( G ) of Banach spaces for each 1 ≤ p ≤ ∞ called coorbit spaces .Each space C o p ( G ) contains the elements f ∈ H π such that W g ( f ) has a certain decay (de-pending on p ) as a function on the group G . To make the definition of the coorbit spaces C o p ( G ) precise, we will first need to extend the wavelet transform to the distributional setting. • By picking a suitable atom g ∈ H π we can generate any f ∈ C o p ( G ) through the formula f = X i ∈ I c i ( f ) π ( x i ) g, (1.1)where { x i } i ∈ I ⊂ G is a collection of carefully chosen points and ( c i ) i ∈ I are coefficients thatdepend linearly on f . This systematic decomposition is known as an atomic decomposition .Intuitively, we decompose each element f ∈ C o p ( G ) into its atomic parts relative to the chosenatom g ∈ H π . The selection of the points { x i } i ∈ I ⊂ G depends heavily on the structure of G ,giving the theory a geometric flavor.Two classes of coorbit spaces that have appeared prominently in the literature are the (homo-geneous) Besov spaces in classical harmonic analysis and the modulation spaces in time-frequencyanalysis. One can obtain a deeper appreciation for these seemingly different spaces by realizing thatthey are both special cases of the coorbit space machinery. These two examples will be returned totime and time again to illustrate the concepts presented. Existing Literature:
There are sources in the literature that deal with coorbit spaces from asomewhat expository viewpoint. We emphasize three of them as they deserve a special mention: • The Ph.D. thesis [63] of Felix Voigtlaender has been very helpful, especially for technicalaspects of the survey. Although the first chapters of [63] are more advanced than this survey(e.g. they deals with quasi-Banach spaces), it nevertheless introduces all the main ideas in aclear manner. • The book [16] is a collection of survey papers written by various authors. Especially Chapter 2(written by Filippo De Mari and Ernesto De Vito) and Chapter 3 (written by Stephan Dahlke,S¨oren H¨auser, Gabriele Steidl, and Gerd Teschke) have been helpful for comprehending thebasics of coorbit theory. • The paper [10] is mostly an expository account of different aspects of coorbit theory. It isboth well-written and useful, although it assumes more background knowledge from the readerthan we do. A drawback is that [10] has, due to its publication date, no modern examplesand directions in coorbit theory. 2s coorbit theory is a popular topic nowadays, there have been several advances of the theory inthe last five years. Most of these topics are not discussed outside of their respective research papers.It is our belief that the community would benefit from having these results more easily available.We will go through some of the recent developments in Section 3.8 and Chapter 4. In Section 4.4we give references to many recent works on coorbit theory.
Unconventional Topics: • Reproducing Kernel Hilbert Spaces : This is included in Section 2.4 since the wavelet transformautomatically produces reproducing kernel Hilbert spaces, see Proposition 2.30. These repro-ducing kernel Hilbert spaces have received interest recently in [3, 56, 61, 40]. The reproducingkernel approach also illuminates the reproducing formula in Theorem 2.32, which is centralto the theory. It should be noted that reproducing kernel Hilbert spaces are often implicitlypresent in works on coorbit theory. • Large Scale Geometry : We have included certain definitions from large scale geometry inSection 3.6 as this provides a convenient language for discussing discretizations. Large scalegeometry has had little intersection with coorbit theory, except for in [53] where it is utilizedsuccessfully. Both [53] and the papers [2, 4] uses large scale geometry to analyze decompositionspaces , which is a family of spaces that are related to coorbit spaces. We hope that large scalegeometry can provide a conceptual framework that might bring new ideas to the table.We have chosen to omit
Wiener amalgam spaces from the survey. This choice is a difficult one;although Wiener amalgam spaces are a useful tool, they are also a conceptional hurdle for someand not always needed in practical applications of coorbit theory. We refer the reader to [24] andthe survey [47] for more details on Wiener amalgam spaces.
Outline: • Chapter 2:
We introduce locally compact groups and unitary representations in Section 2.1and Section 2.2, respectively. In addition to fixing notation, this allows us to require fewprerequisites from the reader. The wavelet transform is a central player in coorbit theoryand is introduced in Section 2.2. We go through the orthogonality relations for the wavelettransform in Section 2.3. In Section 2.4 we review reproducing kernel Hilbert spaces and showthat such spaces naturally arise when considering the wavelet transform. Finally, we derivethe reproducing and reconstruction formulas for the wavelet transform in Section 2.5. • Chapter 3:
We introduce the integrable setting in Section 3.1 and extend the wavelet trans-form to the distributional level in Section 3.2. This allows us to define the coorbit spaces inSection 3.3 in a rigorous manner. The basic properties of the coorbit spaces are derived in Sec-tion 3.4 with the help of the correspondence principle given in Theorem 3.20. In Section 3.5we discuss weighted coorbit spaces. We show that the coorbit spaces have extraordinarysampling properties in Section 3.6 through a general procedure called atomic decompositions .Terminology borrowed from large scale geometry will be used to make the main result inTheorem 3.40 more transparent. Finally, we discuss Banach frames and a kernel theorem forcoorbit theory in respectively Section 3.7 and Section 3.8. • Chapter 4:
We solidify the results presented in previous chapters by giving non-trivialexamples of the theory. This includes shearlet spaces in signal analysis in Section 4.1, Bergmanspaces in complex analysis in Section 4.2, and coorbit spaces built on nilpotent Lie groups3n Section 4.3. We end in Section 4.4 by giving references to recent developments related toembeddings between coorbit spaces and generalizations of coorbit theory.
Acknowledgments:
I have received advice and concrete suggestions from many researchersthroughout the writing process. I would in particular like to thank Stine Marie Berge, Franz Luef,and Felix Voigtlaender for illuminating discussions and helpful comments. Finally, I would like toexpress my gratitude to everyone who participated in the seminar course I gave on coorbit theoryat the Norwegian University of Science and Technology during the fall of 2020.
We start by giving an overview of preliminary topics, namely locally compact groups, unitaryrepresentations, and basic properties of the (generalized) wavelet transform. Most of this materialis fairly standard, and is mainly collected from the books [34, 42, 16, 36, 21]. We aim for a suitablegenerality and present concrete examples as we go along.
The first order of business is to get acquainted with locally compact groups.
Definition 2.1. A locally compact group is a locally compact Hausdorff topological space G thatis simultaneously a group such that the multiplication and inversion maps( x, y ) xy, x x − , x, y ∈ G, are continuous.For us, the object of main importance on a locally compact group is the left Haar measure:Recall that a Radon measure is Borel measure that is finite on compact sets, inner regular on opensets, and outer regular on all Borel sets. Do not worry if you are rusty on the measure-theoreticnonsense; we will never use these technical conditions explicitly. The important point is that eachlocally compact group G can be equipped with a unique (up to a positive constant) left-invariant Radon measure µ L , that is, µ L satisfies µ L ( xE ) = µ L ( E ) for all x ∈ G and every Borel set E ⊂ G .We call the measure µ L the left Haar measure of the group G . The existence of the left Haarmeasure implies that any locally compact group is canonically equipped with a measure-theoreticsetting.As the terminology indicates, there is also a right Haar measure µ R on any locally compact group.How much the two measures µ L and µ R deviate is captured by the modular function ∆ : G → (0 , ∞ )defined as follows: For x ∈ G the measure µ x ( E ) := µ L ( Ex ) is again a left-invariant Radon measure.Therefore, the uniqueness of the left Haar measure implies the existence of a number ∆( x ) ∈ (0 , ∞ )such that µ x ( E ) = ∆( x ) µ L ( E ) , for every Borel set E ⊂ G . It is straightforward to see that µ L = µ R precisely when ∆ ≡ µ L = µ R are called unimodular . When this is the case,we use the abbreviation µ := µ L = µ R and refer to µ as the Haar measure on the group G . It isclear that commutative locally compact groups are unimodular. Moreover, locally compact groupsthat are either compact or discrete are also unimodular, see [34, Chapter 2.4].4 xample 2.2. The reader has surely seen plenty of locally compact groups previously. Two ele-mentary ones are R n with the usual vector sum and R ∗ := R \ { } with the usual product. On R n , the Haar measure is the Lebesgue measure dx , while on R ∗ the Haar measure is dx/ | x | . Toexemplify the last claim, we see for E = ( r, s ) with s > r > x > µ ( xE ) = Z xsxr dtt = log( xs ) − log( xr ) = log (cid:16) sr (cid:17) = µ ( E ) . Example 2.3.
There are many locally compact groups of interest that are not unimodular. As anexample, we consider the (full) Affine group
Aff = R × R ∗ with the group multiplication( b, a ) · ( b ′ , a ′ ) := ( ab ′ + b, aa ′ ) , ( b, a ) , ( b ′ , a ′ ) ∈ Aff . The group operation models the composition of affine maps, and can equivalently be realized as2 × (cid:18) a b (cid:19) , ( b, a ) ∈ Aff , where the group operation is matrix multiplication. Notice that the group operation is not commu-tative. Moreover, the affine group is not unimodular: The reader can verify that the left and rightHaar measures on Aff are respectively given by µ L ( b, a ) = db daa , µ R ( b, a ) = db da | a | . Remark.
If you find yourself in the situation where you have a locally compact group G but noobvious candidate for a Haar measure, then do not despair; there are several ways of construct-ing the Haar measure on many locally compact groups. We refer the interested reader to [34,Proposition 2.21] for a concrete example.For a locally compact group G, we can form the spaces L p ( G ) for 1 ≤ p < ∞ consisting ofequivalence classes of measurable functions f : G → C such that k f k L p ( G ) := (cid:18)Z G | f ( x ) | p dµ L ( x ) (cid:19) p < ∞ . The case p = ∞ also has the obvious extension from the familiar Euclidean case. For locally compactgroups that are not unimodular, some authors use the notation L p ( G, µ L ) for clarity. However, wewill always consider the left Haar measure, and thus boldly use the abbreviated notation L p ( G ).The spaces L p ( G ) are Banach spaces for all 1 ≤ p ≤ ∞ . Moreover, when p = 2 we even have aHilbert space structure given by the inner product h f, g i L ( G ) := Z G f ( x ) g ( x ) dµ L ( x ) . We have for each y ∈ G the left-translation operator L y given by L y f ( x ) := f ( y − x ) for x ∈ G .The reason for the inverse is so that we have L y ◦ L z = L yz for y, z ∈ G . This detail is importantwhen we study unitary representations in Section 2.2. For similar reasons, we define for each y ∈ G the right-translation operator R y by the formula R y f ( x ) := f ( xy ) for x ∈ G . Definition 2.4.
For f, g ∈ L ( G ) we can form the convolution between f and g given by f ∗ G g ( x ) := Z G f ( y ) g ( y − x ) dµ L ( y ) . R n , the convolution is gener-ally non-commutative. In fact, the convolution is commutative precisely when the group operationon G is commutative [21, Theorem 1.6.4]. Moreover, it follows from [48, Corollary 20.14] that theconvolution inequality k f ∗ G g k L p ( G ) ≤ k f k L ( G ) k g k L p ( G ) is valid for all 1 ≤ p ≤ ∞ , g ∈ L p ( G ), and f ∈ L ( G ). Example 2.5.
A non-commutative group that will be of central importance for us is the (full)Heisenberg group H n . As a set we have H n = R n × R n × R , while the group multiplication is givenby (cid:0) x, ω, t (cid:1) · (cid:0) x ′ , ω ′ , t ′ (cid:1) := (cid:18) x + x ′ , ω + ω ′ , t + t ′ + 12 ( x ′ ω − xω ′ ) (cid:19) . Although what we have described is strictly speaking one group for each dimension n , we collectivelyrefer to these groups as the Heisenberg group for simplicity. In Section 2.3 we will use a differentrealization of the Heisenberg group due to integrability issues. The Heisenberg group is unimodularand the Haar measure on H n is the usual Lebesgue measure on R n +1 . We refer the reader to [49]for an excellent exposition on the ubiquity of the Heisenberg group in harmonic analysis. Example 2.6.
When working with locally compact groups, it is advantageous to have both contin-uous and discrete examples in mind. Most discrete examples arise from letting G be any countablegroup with the discrete topology. Let us briefly consider G = Z to see what the convolution lookslike in this case: The Haar measure on Z is the counting measure. It is common to use the notation l p ( Z ) := L p ( Z ) for all 1 ≤ p ≤ ∞ . Per convention, we use sequence notation a = ( a n ) n ∈ Z with a n := a ( n ) for functions a : Z → C . The convolution between two elements a, b ∈ l ( Z ) is preciselythe well-known Cauchy product of sequences given by( a ∗ Z b ) n := ∞ X m = −∞ a m b n − m . We will now consider unitary representations of locally compact groups. This will give rise tothe (generalized) wavelet transform that we will examine closely. Ultimately, we use the wavelettransform to construct the coorbit spaces in Chapter 3. Given a Hilbert space H we let U ( H ) denotethe group of all unitary operators from H to itself. Definition 2.7.
Let G be a locally compact group and let H π be a Hilbert space. A unitaryrepresentation of G on H π is a group homomorphism π : G → U ( H π ) such that the transformation G ∋ x π ( x ) g ∈ H π (2.1)is continuous for all g ∈ H π .It turns out that the continuity requirement (2.1) is equivalent to the seemingly weaker require-ment that G ∋ x g f ( x ) := h f, π ( x ) g i (2.2)is a continuous function on G for all f, g ∈ H π . The function W g f is called the (generalized) wavelettransform of f with respect to g . Hence W g f : G → C is a continuous function by assumption6henever we have a unitary representation. Moreover, we have that W g f is a bounded function on G since |W g f ( x ) | = |h f, π ( x ) g i| ≤ k f kk π ( x ) g k = k f kk g k , x ∈ G. We often take the view that g ∈ H π is fixed and consider the map W g : H π → C b ( G ) sending f to W g f , where C b ( G ) denotes the set of complex valued continuous functions on G that are bounded.The wavelet transform has a central place in coorbit theory, and much of the theory revolves aroundunderstanding subtle properties of this transformation. Example 2.8.
An example of a unitary representation on any locally compact group G is the leftregular representation L : G → U ( L ( G )) given by L ( x ) f ( y ) := L x f ( y ) = f ( x − y ) , for x, y ∈ G and f ∈ L ( G ). Note that L xy = L x ◦ L y and L − x = L x − . Hence the fact that L x isunitary follows from the computation k L x f k L ( G ) = Z G | L x f ( y ) | dµ L ( y ) = Z G | f ( x − y ) | dµ L ( y ) = Z G | f ( y ) | dµ L ( y ) = k f k L ( G ) . For the continuity assertion (2.1), we refer the reader to [34, Proposition 2.42].
Definition 2.9.
Let π : G → U ( H π ) be a unitary representation of a locally compact group G . • We say that a closed subspace
M ⊂ H π is an invariant subspace if π ( x ) g ∈ M for all g ∈ M and x ∈ G . When this happens, the restriction π | M is a unitary representation of G on M and we call π | M : G → U ( M ) a subrepresentation of π . • If there are no non-trivial (other than { } and H π ) invariant subspaces of H π , then π is called irreducible. Otherwise, we say that π is reducible .For any unitary representation π : G → U ( H π ) we have for f, g ∈ H π and x, y ∈ G that W g ( π ( y ) f )( x ) = h π ( y ) f, π ( x ) g i = h f, π ( y − ) π ( x ) g i = W g ( f )( y − x ) = L y [ W g ( f )] ( x ) . (2.3)The simple calculation (2.3) should not be underestimated; it shows that the wavelet transformgives us a way to relate the representation π and the left regular representation L in Example 2.8.This notion is formalized in the following definition. Definition 2.10.
Let G be a locally compact group and consider two unitary representations π : G → U ( H π ) and τ : G → U ( H τ ). We say that a bounded linear operator T : H π → H τ is an intertwiner between π and τ if T ◦ π ( x ) = τ ( x ) ◦ T for every x ∈ G . If T is additionally a unitaryoperator, then we refer to T as a unitary intertwiner . If a unitary intertwiner exists between π and τ , then π and τ are called equivalent .If we are only considering one unitary representation π : G → U ( H π ), then a bounded linearoperator T : H π → H π satisfying T ◦ π ( x ) = π ( x ) ◦ T is simply referred to as a (unitary) intertwiner of π . We leave it to the reader to verify that if π is an irreducible unitary representation and T isa unitary intertwiner between π and another unitary representation τ , then τ is also irreducible.It is tempting, but slightly premature, to reformulate (2.3) in the following way: The wavelettransform W g is, for any choice of g ∈ H π , an intertwiner between π and the left regular represen-tation L given in Example 2.8. The problem is that in general the wavelet transform W g f is not in L ( G ) as the following example shows. 7 xample 2.11. Consider the left regular representation L : R → U ( L ( R )) on G = R . Then for f, g ∈ L ( R ) and x ∈ R the wavelet transform has the form W g f ( x ) = Z ∞−∞ f ( y ) g ( y − x ) dy = f ∗ ˇ g ( x ) , where ˇ g ( x ) := g ( − x ). The space L ( R ) is not closed under convolution: Let f ( x ) = g ( x ) = F (cid:16) e − ω | ω | − (cid:17) ( x ) , where F denotes the Fourier transform. Then one can check that f, g ∈ L ( R ) and W g f L ( R ).We will in Section 2.3 work with additional assumptions on the representation π and the fixedvector g ∈ H π so that W g f ∈ L ( G ) for all f ∈ H π . In that case, a natural question emerges thatwe will answer in Section 2.3: Q: Is W g a unitary intertwiner between π and some subrepresentation of the leftregular representation L ? Example 2.12.
Let us revisit the Heisenberg group H n in Example 2.5 and describe its irreducibleunitary representations. First of all, we have the family of one-dimensional representations of H n given by χ α,β ( x, ω, t ) := e πi ( αx + βω ) ∈ U ( C ) , α, β ∈ R n , ( x, ω, t ) ∈ H n . The central characters χ α,β are obviously irreducible, unitary, and non-equivalent. We refer thereader to [42, Chapter 9.2] for an explanation of why χ α,β are called the central characters of H n .Let T x and M ω be respectively the translation operator and modulation operator on L ( R n )given by T x f ( y ) := f ( y − x ) , M ω f ( y ) := e πiyω f ( y ) , x, y, ω ∈ R n . (2.4)These operators can be combined to form the Schr¨odinger representation ρ : H n → U ( L ( R n )) givenby ρ ( x, ω, t ) f ( y ) := e πit e πixω T x M ω f ( y ) . (2.5)It can be verified that the Schr¨odinger representation is an irreducible unitary representation of H n , see [42, Theorem 9.2.1]. Moreover, one can generate new non-equivalent irreducible unitaryrepresentations by dilating the Schr¨odinger representation ρ λ ( x, ω, t ) := ρ ( λx, ω, λt ) , λ ∈ R \ { } . And that’s it! The Stone-von Neumann theorem [42, Theorem 9.3.1] states that any irreducibleunitary representation of H n is equivalent to either χ α,β for some α, β ∈ R n or ρ λ for some λ ∈ R \{ } . The following result shows a fundamental relationship between irreducible unitary representa-tions and intertwiners.
Lemma 2.13 (Schur’s Lemma) . Let π : G → U ( H π ) be a unitary representation. Then π isirreducible if and only if every intertwiner of π is a constant multiple of the identity Id H π . We refer the reader to [34, Theorem 3.5] for a proof of Schur’s Lemma. One of the main usesof Schur’s Lemma is showing that certain irreducible representations are impossible. The followingresult illustrates this. 8 orollary 2.14.
Let π : G → U ( H π ) be a unitary representation of a commutative locally compactgroup G . If π is irreducible, then dim( H π ) = 1 .Proof. Notice that for all x, y ∈ G we have π ( x ) π ( y ) = π ( xy ) = π ( yx ) = π ( y ) π ( x ) . Thus π ( x ) ∈ U ( H π ) is in fact a unitary intertwiner of π . Hence Schur’s Lemma implies that π ( x ) = C x · Id H π for all x ∈ G , where C x is a constant dependent on x . However, it is now clearthat any closed subspace of H π is invariant. This can only be the case, under the assumption ofirreducibility, when H π does not have any closed subspaces other than { } and H π .Let us try to construct an invariant subspace of a unitary representation π : G → U ( H π ). Fix anon-zero vector g ∈ H π and form the subspace M g := span { π ( x ) g : x ∈ G } ⊂ H π . Notice that M g is a closed subspace of H π that is non-trivial since g = π ( e ) g ∈ M g , where e ∈ G is the identity element of G . Moreover, M g is clearly invariant under the action of π . We call M g the cyclic subspace generated by g ∈ H π . If M g = H π , then the vector g is said to be cyclic . Ifthis is not the case, then the representation π is reducible as M g would be a non-trivial invariantsubspace. Conversely, assume that every non-zero vector g ∈ H π is cyclic and let M ⊂ H π be anon-trivial invariant subspace. Fix a non-zero g ∈ M and notice that M g ⊂ M . Since g is cyclicthis forces M = H π so that π is irreducible. We summarize this discussion for later reference in thefollowing proposition. Proposition 2.15.
A unitary representation π : G → U ( H π ) is irreducible precisely when everynon-zero vector g ∈ H π is cyclic. The following result shows that cyclic vectors are of central importance for the wavelet transform.
Lemma 2.16.
Consider a unitary representation π : G → U ( H π ) and fix a non-zero vector g ∈ H π .The wavelet transform W g : H π → C b ( G ) is injective if and only if g is a cyclic vector.Proof. Assume by contradiction that g is a cyclic vector and W g is not injective. Pick f ∈ H π \ { } such that W g f is the zero function on G , that is, W g f ( x ) = h f, π ( x ) g i = 0 , for all x ∈ G . This implies that f is orthogonal to the cyclic subspace M g . In particular, M g = H π and we have a contradiction. Conversely, assume that g is not cyclic so that M g = H π . Bypicking f ∈ M ⊥ g \ { } we have that h f, π ( x ) g i = 0 for all x ∈ G . Hence W g : H π → C b ( G ) is notinjective. We want to examine the wavelet transform W given in (2.2) in more detail. It is instructive to lookat a concrete example first to see what we might expect. Example 2.17.
Let us consider the Schr¨odinger representation ρ of the Heisenberg group H n givenin (2.5). The wavelet transform corresponding to this representation is given by W g f ( x, ω, t ) = h f, ρ ( x, ω, t ) g i = e − πit e − πixω h f, T x M ω g i = e − πit e πixω h f, M ω T x g i , (2.6)9or f, g ∈ L ( R n ). We can recognize the term h f, M ω T x g i as the short-time Fourier transform (STFT), which is usually denoted by V g f ( x, ω ) := h f, M ω T x g i = Z R n f ( t ) g ( t − x ) e − πitω dt. Hence the wavelet transform for the Schr¨odinger representation is, up to a phase factor, the short-time Fourier transform. The STFT satisfies two important properties:
Orthogonality:
For f , f , g , g ∈ L ( R n ) we have the orthogonality relation h V g f , V g f i L ( R n ) = h f , f i L ( R n ) h g , g i L ( R n ) . (2.7) Reconstruction:
Fix g ∈ L ( R n ) with k g k L ( R n ) = 1. Given any f ∈ L ( R n ), we can reconstruct f from V g f through the formula h f, h i L ( R n ) = Z R n V g f ( x, ω ) V g h ( x, ω ) dx dω, (2.8)for any h ∈ L ( R n ).The proofs can be found in [42, Theorem 3.2.1] and [42, Corollary 3.2.3], respectively.We postpone discussing the reconstruction property (2.8) to Section 2.5. It turns out that theSTFT case is a best case scenario; not all generalized wavelet transforms exhibit such a simpleorthogonality relation. From (2.7) we see that V g : L ( R n ) → L ( R n ) is an isometry for anynormalized g ∈ L ( R n ). Generalizing this observation, we would like to answer the followingquestion in this section: Q: Under which conditions on a general unitary representation π : G → U ( H π )and a non-zero vector g ∈ H π can we ensure that the generalized wavelet transform W g : H π → L ( G ) is an isometry?Notice that this question is precisely the same as the question we asked in Section 2.2 regardingwhether W g is a unitary intertwiner between π and a subrepresentation of the left regular represen-tation. Given a unitary representation π : G → U ( H π ) we first of all need that W g is injective. ByProposition 2.15 and Lemma 2.16 this will be satisfied for all non-zero vectors g ∈ H π whenever π is irreducible. Henceforth we will require that π is irreducible. Secondly, we need a condition on g to ensure that W g f ∈ L ( G ) for all f ∈ H π . Definition 2.18.
Let π : G → U ( H π ) be an irreducible unitary representation. We say that anon-zero vector g ∈ H π is square integrable if W g g ∈ L ( G ). Explicitly, we require that Z G |h g, π ( x ) g i| dµ L ( x ) < ∞ . The representation π is said to be square integrable if there exists at least one square integrablevector for π . Remark.
Pay attention to the fact that a square integrable representation π of a locally compactgroup G is both unitary and irreducible by definition. These assumptions are implicit whenever wesay that a representation π : G → U ( H π ) is square integrable. A stronger requirement one couldimpose is for a non-zero vector g to be integrable in the sense that W g g ∈ L ( G ). It follows fromthe inclusion L ( G ) ∩ L ∞ ( G ) ⊂ L ( G ) that every integrable vector is square integrable. We willreturn to this more stringent condition in Chapter 3.10 xample 2.19. An irreducible unitary representation is not automatically square integrable: Con-sider the trivial representation π : G → U ( C ) given by π ( x ) = Id C for all x ∈ G . Then for z ∈ C \{ } we have Z G |h z, π ( x ) z i| dµ L ( x ) = Z G |h z, z i| dµ L ( x ) = | z | µ L ( G ) . Hence the trivial representation of G is square integrable if and only if µ L ( G ) < ∞ . This in turnhappens if and only if G is compact by [21, Proposition 1.4.5]. Since the wavelet transform iscontinuous, it is clear that any irreducible unitary representation of a compact group is automaticallysquare integrable. In fact, it is not terribly difficult to show that a locally compact group G has asquare integrable representation on a finite dimensional vector space if and only if G is compact,see [60, Proposition 16.4]. Example 2.20.
The wavelet transform (2.6) for the Schr¨odinger representation is not square in-tegrable. This is due to the last component { } × { } × R being only present in the phase factors.Notice that ρ ( x, ω, t ) = Id L ( R n ) precisely whenever ( x, ω, t ) = (0 , , n ) for n ∈ Z . Hence we canconsider the quotient group H nr := H n / ker( ρ ) ≃ R n × R n × T with the Haar measure dx dω dτ andthe product (cid:0) x, ω, e πiτ (cid:1) · (cid:16) x ′ , ω ′ , e πiτ ′ (cid:17) := (cid:16) x + x ′ , ω + ω ′ , e πi ( τ + τ ′ ) e πi ( x ′ ω − xω ′ ) (cid:17) , for x, x ′ , ω, ω ′ ∈ R n and τ, τ ′ ∈ R . The group H nr is called the reduced Heisenberg group .The Schr¨odinger representation ρ : H n → U ( L ( R n )) descends to an irreducible unitary repre-sentation ρ r : H nr → U ( L ( R n )) given by ρ r (cid:0) x, ω, e πiτ (cid:1) f ( y ) = e πiτ e πixω T x M ω f ( y ) , (cid:0) x, ω, e πiτ (cid:1) ∈ H nr , where T x and M ω are given in (2.4). Although sloppy, it is common to refer to ρ r as the Schr¨odingerrepresentation as well. In contrast with ρ , the representation ρ r is square integrable: For any non-zero g ∈ L ( R n ) we have kW g g k L ( H nr ) = Z Z R n Z R n | V g g ( x, ω ) | dx dω dτ = k V g g k L ( R n ) = k g k L ( R n ) , (2.9)where we used the orthogonality relation (2.7) of the STFT. Hence the map W g is an isometry from L ( R n ) to L ( H nr ) when k g k L ( R n ) = 1.At first glance, the condition that g ∈ H π is square integrable seems slightly weaker than therequirement desired, namely that W g f ∈ L ( G ) for all f ∈ H π . However, it turns out that they arein fact equivalent. Proposition 2.21.
Let π : G → U ( H π ) be a square integrable representation with a square integrablevector g ∈ H π . Then W g f ∈ L ( G ) for all f ∈ H π .Proof. Consider the subspace H g ⊂ H π consisting of those f ∈ H π such that W g f ∈ L ( G ). Then H g is a non-trivial subspace since g ∈ H g . The fact that H g is closed in H π is rather tricky, andwe refer the reader to [67, Lemma 6.3] for the argument. Notice that H g is an invariant subspacesince (2.3) shows that W g π ( x ) f = L x W g f, f ∈ H g , x ∈ G. By irreducibility, we have H g = H π and the result follows.11 emark. There are several ways of characterizing square integrable representations that we will notemphasize. One of the more elegant formulations [22, Theorem 2] states that an irreducible unitaryrepresentation is square integrable precisely when it is equivalent to a subrepresentation of the leftregular representation. In the literature, e.g. [36], such representations are sometimes referred to as discrete series representations .The next result gives a complete answer to how the orthogonality relation (2.7) generalizes toarbitrary square integrable representations.
Theorem 2.22 (Duflo-Moore Theorem) . Let π : G → U ( H π ) be a square integrable representation.There exists a unique self-adjoint, positive, densely defined operator C π : D ( C π ) ⊂ H π → H π witha densely defined inverse such that: • A non-zero element g ∈ H π is square integrable if and only if g ∈ D ( C π ) . • For g , g ∈ D ( C π ) and f , f ∈ H π we have the orthogonality relation hW g f , W g f i L ( G ) = h f , f i H π h C π g , C π g i H π . (2.10) • The operator C π is injective and satisfies the invariance relation π ( x ) C π = p ∆( x ) C π π ( x ) , (2.11) for all x ∈ G where ∆ denotes the modular function on G . For readers interested in the details of this remarkable result, we recommend reading the ap-pendix in [45, Chapter 2.4] as well as the original paper [22]. We will refer to the operator C π inTheorem 2.22 as the Duflo-Moore operator corresponding to the square integrable representation π : G → U ( H π ). For our purposes, we record the following consequence: The map W g : H π → L ( G )is an isometry if and only if g ∈ H π is in the domain of the Duflo-Moore operator C π and satisfiesthe admissibility condition k C π g k H π = 1 . An element g ∈ H π that satisfies these conditions is said to be admissible . Notice that any squareintegrable vector can be normalized to become admissible. Corollary 2.23.
Let π : G → U ( H π ) be a square integrable representation of a unimodular group G . Then the Duflo-Moore operator C π is defined on the whole H π and satisfies C π = c π · Id H π forsome c π > . In particular, every non-zero vector g ∈ H π is square integrable.Proof. By looking at the invariance relation (2.11) when ∆( x ) = 1 for all x ∈ G , we see that C π isa (densely defined) intertwiner of the representation π . This is only possible when C π = c π · Id H π for some constant c π ∈ C due to a generalization of Schur’s Lemma, see [21, Proposition 12.2.2].The constant c π necessarily has to be positive since C π is a positive operator. Remark.
We would like to point out that a converse statement to Corollary 2.23 is also valid: If π : G → U ( H π ) is a square integrable representation such that the Duflo-Moore operator C π isdefined on the whole of H π , then the group G is necessarily unimodular. To see this, one usesthe invariance relation (2.11) together with the general fact that the modular function ∆ is eitheridentically one or unbounded. For the last property, it suffices that ∆ : G → (0 , ∞ ) is a grouphomomorphism by [34, Proposition 2.24]. 12 xample 2.24. Let us quickly verify that the Schr¨odinger representation ρ r does indeed fit withinthis framework. We have previously mentioned that the reduced Heisenberg group H nr is unimodular.Hence Corollary 2.23 implies that the Duflo-Moore operator C ρ r corresponding to ρ r is simply aconstant times the identity. We can gauge from (2.9) that C ρ r = Id L ( R n ) . Hence a function g ∈ L ( R n ) is admissible for the Schr¨odinger representation precisely when k g k L ( R n ) = 1. Example 2.25.
Let π : G → U ( H π ) be an irreducible unitary representation of a compact group G .From Peter-Weyl theory, see e.g. [21, Theorem 7.3.2], it follows that H π has to be finite dimensional.Moreover, any non-zero vector g ∈ H π is square integrable since W g g is a continuous function onthe compact space G . Thus the Duflo-Moore operator satisfies C π = c π · Id H π for some c π >
0. So,what is the constant c π ? It follows from [21, Example 12.2.7] that we have the elegant formula c π = 1 p dim( H π ) . Example 2.26.
Let us demonstrate how Theorem 2.22 can simplify concrete settings: Considertwo normalized vectors x, y ∈ R n and a rotation R ∈ SO ( n ). The quantity |h y, Rx i| measuresthe square deviation from Rx and y being orthogonal. What is the average of such orthogonalitydeviations when the normalized vectors x, y ∈ R n are fixed and R ∈ SO ( n ) is allowed to vary?Unwinding the question, we are asking for the value Z SO ( n ) |h y, Rx i| dµ ( R ) , x, y ∈ R n , k x k = k y k = 1 . When n = 2 the answer should be 1 / R ∈ SO (2) can be written as R = R θ for θ ∈ [0 , π ) with R θ := (cid:18) cos( θ ) − sin( θ )sin( θ ) cos( θ ) (cid:19) . Is there a more satisfactory approach that works for all n ≥
2? Look closely, there is nothingup my sleeve: Consider the obvious representation π : SO ( n ) → U ( R n ) given by π ( R ) x := R · x for R ∈ SO ( n ) and x ∈ R n . Then π is easily seen to be square integrable. We can by Theorem 2.22and Example 2.25 write Z SO ( n ) |h y, Rx i| dµ ( R ) = Z SO ( n ) |W x y ( R ) | dµ ( R ) = h y, y ih C π x, C π x i = 1 n . (2.12)In words, the formula (2.12) expresses the fact that in higher dimensions, two random normalizedvectors are more likely to be orthogonal to each other; there are simply more ways to be orthogonalin higher dimensions.We would like to end this section with an example of a square integrable representation of anon-unimodular group. Although somewhat lengthy, we encourage the fatigued reader to soldieron through the next example as most of the theory we have developed is present in some way. Example 2.27.
In this example we examine a unitary representation of the affine group Aff givenin Example 2.3. We have a family of dilation operators D a on L ( R ) for a ∈ R ∗ given by D a f ( x ) := 1 p | a | f (cid:16) xa (cid:17) , f ∈ L ( R ) . (2.13)13ogether with the translation operator T b in (2.4) we obtain a unitary representation of the affinegroup π : Aff → U ( L ( R )) given by π ( b, a ) f ( x ) := T b D a f ( x ) = 1 p | a | f (cid:18) x − ba (cid:19) , ( b, a ) ∈ Aff . (2.14)It is common to refer to π as the wavelet representation . To see that a unitary representation isirreducible, it can often be a good strategy to jump straight to checking when it is square integrable.For the wavelet representation, a formal computation using the Fourier transform shows that Z Aff |h f, π ( b, a ) g i| db daa = (cid:18)Z R |F ( f )( b ) | db (cid:19) (cid:18)Z R ∗ |F ( g )( a ) | | a | da (cid:19) , (2.15)for any f, g ∈ L ( R ). We refer the reader to [16, Example 2.48] for details of the computationabove. The right-hand side of (2.15) is always non-zero as long as we choose f, g to be non-zeroelements in L ( R ). Hence g is a cyclic vector for all non-zero g ∈ L ( R ). This implies that thewavelet representation π is irreducible by Proposition 2.15.Which non-zero vectors g ∈ L ( R ) are square integrable? From (2.15), we see that we need g tosatisfy the condition Z R ∗ |F ( g )( a ) | | a | da < ∞ . (2.16)The condition (2.16) is sometimes called the Calder´on condition or the wavelet condition . It is clearfrom (2.15) and the uniqueness statement of Theorem 2.22 that the Duflo-Moore operator C π is theFourier multiplier given by C π g = F − p | a | F ( g )( a ) ! , g ∈ D ( C π ) . We know that g ∈ D ( C π ) is admissible if and only if k C π g k L ( R ) = 1. Hence g ∈ L ( R ) is admissiblefor the wavelet representation if and only if Z R ∗ |F ( g )( a ) | | a | da = 1 . (2.17)Elements in L ( R ) that satisfy (2.17) are sometimes called admissible wavelets in the literature.The wavelet transform for the wavelet representation is given explicitly by W g f ( b, a ) = h f, T b D a g i = 1 p | a | Z R f ( x ) g (cid:18) x − ba (cid:19) dx, (2.18)where ( b, a ) ∈ Aff and f, g ∈ L ( R ). This is precisely the continuous wavelet transform in waveletanalysis, see e.g. [20, Chapter 2]. In fact, this example is the motivation for the terminology (generalized) wavelet transform . If g ∈ L ( R ) is an admissible wavelet and f , f ∈ L ( R ) arearbitrary, then Theorem 2.22 implies that we have the orthogonality relation Z Aff W g f ( b, a ) W g f ( b, a ) db daa = Z R f ( x ) f ( x ) dx. .4 Reproducing Kernel Hilbert Spaces In this section we define reproducing kernel Hilbert spaces and show that they naturally occur inthe setting of generalized wavelet transforms. We believe that reproducing kernel Hilbert spacescan illuminate the theory and make results such as Theorem 2.32 in Section 2.5 more transparent.Although the theory of reproducing kernel Hilbert spaces is often implicit in works on coorbit theory,it is seldom written out in detail.
Definition 2.28.
Let X be a set and let H be a Hilbert space consisting of functions f : X → C . Wesay that H is a reproducing kernel Hilbert space if the evaluation functionals { E x } x ∈ X are bounded,where E x ( f ) := f ( x ) , f ∈ H . If the evaluation functionals { E x } x ∈ X are uniformly bounded, then we refer to H as a uniformreproducing kernel Hilbert space .Given a reproducing kernel Hilbert space H , we have by the Riesz representation theorem thatfor each x ∈ X there is a unique element k x ∈ H such that f ( x ) = h f, k x i , f ∈ H . We refer to k x as the reproducing kernel for the point x ∈ X . Since k x is again a function on X , wecan evaluate k x ( y ) for y ∈ X and obtain k x ( y ) = h k x , k y i = k y ( x ). The function K : X × X → C given by K ( x, y ) = h k y , k x i is called the reproducing kernel for H . Example 2.29.
Consider the
Paley-Wiener space
P W A for a fixed A > f ∈ L ( R ) such that supp( F ( f )) ⊂ [ − A, A ], where F denotes the Fourier transform. This spaceplays a major role in sampling theory and classical harmonic analysis. The elements in P W A areactually smooth functions since their Fourier transforms have compact support. Moreover, thespace P W A is a Hilbert spaces under the inner-product h f, g i P W A := hF ( f ) , F ( g ) i L [ − A,A ] . To see that the evaluation functionals { E x } x ∈ R are bounded, we compute for f ∈ P W A that | E x ( f ) | = | f ( x ) | = (cid:12)(cid:12) F − ( F ( f )) ( x ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z A − A F ( f )( ω ) e πixω dω (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18)Z A − A |F ( f )( ω ) | dω (cid:19) (cid:18)Z A − A dω (cid:19) = √ A · k f k P W A . Since
A >
P W A is a uniform reproducing kernel Hilbert space. To findthe reproducing kernel K A : R × R → C , notice that f ( x ) = h f, k x i P W A = Z A − A F ( f )( ω ) F ( k x )( ω ) dω. In view of the Fourier inversion f = F − ( F ( f )), it follows that F ( k x )( ω ) = e − πixω . Hence K A ( x, y ) = k x ( y ) = F − ( e − πix · )( y ) = ( π sin(2 πA ( x − y )) x − y , if x = y A, if x = y .
15 useful feature of reproducing kernel Hilbert spaces is that convergence in norm implies point-wise convergence. To see this, let f n , f ∈ H and assume k f n − f k →
0. Then | f n ( x ) − f ( x ) | = |h f n − f, k x i| ≤ k f n − f kk k x k = k f n − f kk E x k → . (2.19)If H in addition is a uniform reproducing kernel Hilbert space, then (2.19) shows that convergencein norm implies uniform convergence. The reader can consult [58] for more examples and propertiesof general reproducing kernel Hilbert spaces.We now return to the setting of square integrable representations π : G → U ( H π ) to illustratehow they naturally give rise to reproducing kernel Hilbert spaces. Pick an admissible vector g ∈ H π so that W g : H π → L ( G ) is an isometry. We will consider the image space W g ( H π ) ⊂ L ( G ) . Notice that, since W g is an isometry, we have W ∗ g ◦ W g = Id H π and W g ◦ W ∗ g (cid:12)(cid:12)(cid:12) W g ( H π ) = Id W g ( H π ) . (2.20) Proposition 2.30.
Let π : G → U ( H π ) be a square integrable representation with an admissiblevector g ∈ H π . The space W g ( H π ) is a uniform reproducing kernel Hilbert space with reproducingkernel K g ( x, y ) = W g g ( y − x ) , x, y ∈ G. Proof.
The admissibility of g ∈ H π ensures that W g ( H π ) is a closed subspace of L ( G ). Thus W g ( H π ) is a Hilbert space with the norm kW g f k W g ( H π ) := kW g f k L ( G ) = k f k H π , f ∈ H π . For F ∈ W g ( H π ) and x ∈ G we can thus write F ( x ) = W g (cid:0) W ∗ g F (cid:1) ( x ) = (cid:10) W ∗ g F, π ( x ) g (cid:11) = h F, W g ( π ( x ) g ) i . Since W g ( π ( x ) g ) ∈ W g ( H π ) we have that W g ( H π ) is a reproducing kernel Hilbert space. Thereproducing kernel K g : G × G → C is given by K g ( x, y ) = hW g ( π ( y ) g ) , W g ( π ( x ) g ) i = h π ( y ) g, π ( x ) g i = W g ( π ( y ) g )( x ) = W g g ( y − x ) . If E x is the evaluation functional for the point x ∈ G then k E x k W g ( H π ) ∗ = k k x k W g ( H π ) = kW g ( π ( x ) g ) k W g ( H π ) = k π ( x ) g k H π = k g k H π . It follows that W g ( H π ) is a uniform reproducing kernel Hilbert space since we have fixed g .For a locally compact group G , we say that an element S ∈ L ( G ) is self-adjoint convolutionidempotent if S ( x ) = S ( x − ) for all x ∈ G and S ∗ G S = S . It will follow from Theorem 2.32 thatthe element W g g is self-adjoint convolution idempotent whenever g ∈ H π is admissible. A converseto this statement can be found in [36, Proposition 2.38]. In [36, Theorem 2.45] the followinggeneralization of a classical result of Wilczok [66] is derived. Proposition 2.31.
Let G be a locally compact group that is connected and non-compact. Consider asquare integrable representation π : G → U ( H π ) and fix an admissible vector g ∈ H π . If F ∈ W g ( H π ) is supported on a set of finite Haar measure, then F ≡ .Remark. The reader can consult [36, Chapter 2.5] for more interesting results regarding self-adjointconvolution idempotents. We refer the reader to [3, 40] for further properties of the spaces W g ( H π ).16 .5 The Reproducing and Reconstruction Formulas We end this chapter by providing two important results that tie up loose ends. Firstly, we provethe reproducing formula in Theorem 2.32. This result has a simple interpretation in the language ofreproducing kernel Hilbert spaces. Secondly, we generalize the reconstruction formula for the STFTin (2.8) to square integrable representations in Corollary 2.34. Both of these results have short andelegant proofs that build on the theory developed so far.
Theorem 2.32 (Reproducing Formula) . Let π : G → U ( H π ) be a square integrable representationand fix an admissible vector g ∈ H π . Then W g ◦ W ∗ g is the projection from L ( G ) to W g ( H π ) andhas the explicit form W g ◦ W ∗ g ( F ) = F ∗ G W g g, F ∈ L ( G ) . In particular, for F ∈ W g ( H π ) we have F = F ∗ G W g g. (2.21) Proof.
The map W g : H π → L ( G ) is an isometry since g ∈ H π is admissible. Hence W g ◦ W ∗ g is theprojection from L ( G ) to W g ( H π ). For x ∈ G and F ∈ L ( G ), an initial computation using (2.3)shows that W g (cid:0) W ∗ g F (cid:1) ( x ) = hW ∗ g ( F ) , π ( x ) g i = h F, W g ( π ( x ) g ) i = h F, L x W g g i . Since W g g ( x ) = W g g ( x − ) we end up with h F, L x W g g i = Z G F ( y ) W g g ( x − y ) dµ L ( y ) = Z G F ( y ) W g g ( y − x ) dµ L ( y ) = ( F ∗ G W g g )( x ) . Remark.
The special case (2.21) motivates the name reproducing formula , as we can reproduce thevalues of F ∈ W g ( H π ) by convolving F with W g g ∈ W g ( H π ). Notice that W g g is precisely thereproducing kernel k e for the identity element e ∈ G . Hence (2.21) shows that the reproducingkernel k e is a (right) identity for W g ( H π ) with respect to the convolution product. The fact thatthe wavelet transform W g for any admissible g ∈ H π is an isomorphism W g : H π ∼ −→ W g ( H π ) = (cid:8) F ∈ L ( G ) : F = F ∗ G W g g (cid:9) is a special case of the correspondence principle in Theorem 3.20.We now take a brief detour to weak integrals so that uninitiated readers will be less squeamishwhen encountering expressions on the form (2.23). Let Φ : G → H be a continuous function from alocally compact group G to a Hilbert space H . We need to make sense of Z G Φ( x ) dµ L ( x ) (2.22)as an element in H . This can be done under a mild additional requirement. Specifically, we requirethat the linear functional on H given by f Z G h Φ( x ) , f i dµ L ( x )is well-defined and bounded. Under this assumption, the Riesz representation theorem implies theexistence of an element in H denoted by (2.22) such that (cid:28)Z G Φ( x ) dµ L ( x ) , f (cid:29) = Z G h Φ( x ) , f i dµ L ( x ) , for every f ∈ H . We refer to the element (2.22) as the weak integral of the function Φ.17 roposition 2.33. Let π : G → U ( H π ) be a square integrable representation and fix an admissiblevector g ∈ H π . Then for F ∈ L ( G ) we can represent W ∗ g ( F ) as the weak integral W ∗ g ( F ) = Z G F ( x ) π ( x ) g dµ L ( x ) . (2.23) Proof.
Notice that Φ F : G → H π given by Φ F ( x ) := F ( x ) π ( x ) g for F ∈ L ( G ) satisfies the requiredproperties for a weak integral due to the assumed continuity of π and the estimate (cid:12)(cid:12)(cid:12)(cid:12)Z G h F ( x ) π ( x ) g, f i dµ L ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z G |h F ( x ) π ( x ) g, f i| dµ L ( x )= Z G | F ( x ) | · |W g f ( x ) | dµ L ( x ) ≤ k F k L ( G ) kW g f k L ( G ) = k F k L ( G ) k f k H π , for f ∈ H π . The claim hence follows from the computation hW ∗ g ( F ) , f i H π = h F, W g f i W g ( H π ) = Z G F ( x ) W g f ( x ) dµ L ( x ) = Z G h F ( x ) π ( x ) g, f i dµ L ( x ) . By combining Proposition 2.33 with (2.20) we obtain the following generalization of the recon-struction formula for the STFT given in (2.8).
Corollary 2.34 (Reconstruction Formula) . Let π : G → U ( H π ) be a square integrable representa-tion and fix an admissible vector g ∈ H π . We can represent any f ∈ H π as the weak integral f = W ∗ g ( W g f ) = Z G W g f ( x ) π ( x ) g dµ L ( x ) . Hence we have for any h ∈ H π that h f, h i = Z G W g f ( x ) W g h ( x ) dµ L ( x ) . Example 2.35.
For the wavelet representation given in Example 2.27, the reconstruction formulain Corollary 2.34 takes the form f = Z Aff W g f ( b, a ) T b D a g db daa , for f ∈ L ( R ) arbitrary and g ∈ L ( R ) an admissible wavelet. In this chapter we will define the coorbit spaces and derive their basic properties. The coorbitspaces consist of elements η such that the wavelet transform W g η has suitable decay as a functionon the group G . However, the elements η will not be picked from H π , but rather from a largerdistributional space. The aim of the first two sections in this chapter is to make this notion precise.Once this is ready, we will define coorbit spaces without weights in Section 3.3. The weightedversions will be introduced in Section 3.5 so that we can initially introduce coorbit spaces with18inimal technicalities. Although this is an uncommon approach in the literature, we believe thatwhat this approach lacks in efficiency is made up for by increased clarity. In Section 3.6 we showthat the coorbit spaces can be discretized in a way that reflects the geometry of the underlyinggroup. Finally, we discuss Banach frames and kernel theorems for coorbit spaces respectively inSection 3.7 and Section 3.8. Restriction to σ -compact groups: For some results in this chapter, we will need that thelocally compact group G is σ -compact , that is, there exists a sequence of compact sets ( K n ) n ∈ N with K n ⊂ G such that ∪ n ∈ N K n = G . Rather than explicitly requiring this at individual points inthe exposition, we henceforth restrict our attention to σ -compact groups. Whenever we refer to arepresentation π : G → U ( H π ), it is from now on implicitly assumed that G is a σ -compact locallycompact group. We remark that σ -compactness for locally compact groups is a mild condition: Notonly is any second countable or connected locally compact group (e.g. any Lie group) σ -compact,but by [34, Proposition 2.4] we can always find a subgroup of a locally compact group that is open,closed, and σ -compact. In this section we will go from the square integrable setting to the integrable setting. An irreducibleunitary representation π : G → U ( H π ) is said to be integrable if there exists an integrable vector , thatis, if there is a non-zero vector g ∈ H π such that W g g ∈ L ( G ). Notice that π is then automaticallysquare integrable since W g g ∈ L ( G ) ∩ L ∞ ( G ) ⊂ L ( G ). We use the notation A := (cid:8) g ∈ H π : W g g ∈ L ( G ) (cid:9) . The set A is sometimes called the analyzing vectors in the literature [28]. Notice that A containsall the integrable vectors as well as the zero vector.From now on, we will require that A is non-trivial, that is, we require that the representation π is integrable. Given an integrable vector g ∈ A \ { } , we can define the corresponding space of testvectors H g := (cid:8) f ∈ H π : W g f ∈ L ( G ) (cid:9) . The terminology “test vectors” is not standard, although it has been used in [1, 63]. We will explainin Section 3.2 why this terminology is suitable.The space H g can be equipped with the norm k f k H g := kW g f k L ( G ) , f ∈ H g . To see that this is a norm and not just a seminorm we assume that kW g f k L ( G ) = 0 for some f ∈ H g . Then W g f is zero almost everywhere as a function on G . This implies that W g f representsthe zero equivalence class in L ( G ). The injectivity of W g : H π → L ( G ) ensured by Proposition2.15 and Lemma 2.16 gives that f = 0 as an element in H π , and hence also as an element in H g . Proposition 3.1.
Let π : G → U ( H π ) be an integrable representation and fix an integrable vector g ∈ A \ { } . The restriction π | H g acts by isometries on the set of test vectors H g . Furthermore, thespace of test vectors H g is dense in H π .Proof. It is clear that H g is a linear subspace of H π . Moreover, for f ∈ H g we have for x ∈ G that k π ( x ) f k H g = kW g π ( x ) f k L ( G ) = k L x W g f k L ( G ) = kW g f k L ( G ) = k f k H g . Hence the closure of H g in the norm on H π is a non-trivial closed subspace of H π where π acts byisometries. The irreducibility of π implies that H g is a dense subspace of H π .19 emark. It is tempting, in light of Proposition 2.21, to attempt to show that H g is closed in H π .Then Proposition 3.1 would show that H g = H π . However, this is generally false and we will givea concrete counterexample in Example 3.7. In fact, it will be clear from Section 3.3 that coorbittheory is not very interesting whenever H g = H π . Proposition 3.2.
Let π : G → U ( H π ) be an integrable representation. Then for any integrablevector g ∈ A \ { } the test vectors H g form a Banach space that is continuously embedded into H π .Proof. We begin by showing that the space H g is continuously embedded into H π . For an element f ∈ H g , we have by the orthogonality relations in (2.10) that k C π g k H π k f k H π = kW g f k L ( G ) = Z G |h f, π ( x ) g i||W g f ( x ) | dµ L ( x ) ≤ Z G k f k H π k π ( x ) g k H π |W g f ( x ) | dµ L ( x )= k f k H π k g k H π kW g f k L ( G ) . Since C π g = 0 due to the injectivity of C π , we can rearrange and obtain k f k H π ≤ k g k H π k C π g k H π kW g f k L ( G ) = k g k H π k C π g k H π k f k H g . (3.1)Let us now show that H g is complete. Assume that { f n } n ∈ N is a Cauchy-sequence in H g . Bydefinition, this means that for every ǫ > N ∈ N such that for n, m ≥ N we have kW g f n − W g f m k L ( G ) = k f n − f m k H g < ǫ. Now, by completeness of L ( G ), the sequence W g f n converges to an element F ∈ L ( G ). Moreover,we see from (3.1) that there exists an element f ∈ H π such that f n converges to f in the normon H π . Hence by the continuity of W g as a transformation from H π to L ( G ), the sequence W g f n converges to W g f in the L ( G )-norm. However, since W g ( H π ) is a reproducing kernel Hilbert space,we know that the convergence W g f n → W g f is also valid pointwise. This forces F = W g f . Hence f ∈ H g and f n → f in the H g -norm, showing that H g is complete.The main goal of this section is to show in Theorem 3.5 that the set of test vectors H g does notdepend on the choice of integrable vector g ∈ A \ { } . To do this, we first need two preliminaryresults given in Lemma 3.3 and Lemma 3.4 regarding the Duflo-Moore operator C π and integrablevectors. These technicalities are somewhat neglected in the original sources [29, 30, 28] on coorbittheory. To our knowledge, this was first put on rigorous footing in [63, Lemma 2.4.5]. Lemma 3.3.
Let π : G → U ( H π ) be an integrable representation. Then C π ( A ) ⊂ D ( C π ) , where D ( C π ) denotes the domain of the Duflo-Moore operator.Proof. Due to the self-adjointness of C π , it suffices to show that C π ( g ) ∈ D ( C ∗ π ) for all g ∈ A . Toshow this, we prove that the linear functional on D ( C π ) given by f C π f, C π g i H π = h f, C ∗ π C π g i H π
20s bounded. For g = 0 the boundedness clearly holds. For g = 0 the claim follows from theorthogonality relation (2.10) since |h C π f, C π g i H π | = k g k − H π |hW g g, W f g i L ( G ) |≤ k g k − H π kW g g k L ( G ) kW f g k L ∞ ( G ) ≤ (cid:16) k g k − H π k g k H g (cid:17) · k f k H π . Lemma 3.4.
Let π : G → U ( H π ) be an integrable representation and fix two integrable vectors g , g ∈ A \ { } . Then there exists an integrable vector g ∈ A \ { } such that h C π g, C π g i i 6 = 0 , i = 1 , . Proof. If h C π g , C π g i 6 = 0, then we can simply take g = g . The injectivity of the Duflo-Mooreoperator C π ensures that h C π g , C π g i 6 = 0. Hence we are left with the case h C π g , C π g i = 0.We point out that Lemma 3.3 allows us to consider C π ( C π ( g )). Notice that neither C π g nor C π ( C π ( g )) can be zero due to the injectivity of C π . Since the collection { π ( x ) g } x ∈ G is dense in H π , there exists some fixed x ∈ G such that0 = h π ( x ) g , C π ( C π ( g )) i = h C π ( π ( x ) g ) , C π g i , where we used that C π is self-adjoint. The desired element we need will be of the form g := g + ǫ · π ( x ) g for some ǫ > g ∈ A \ { } for every ǫ >
0. This follows from the calculation W g g = W g g + ǫ · (cid:0) W g π ( x ) g + W π ( x ) g g (cid:1) + ǫ · W π ( x ) g π ( x ) g = W g g + ǫ · ( L x W g g + R x W g g ) + ǫ · L x R x W g g , together with the fact that L ( G ) is both left-invariant and right-invariant. To see that g satisfiesthe required properties, we first have that h C π g, C π g i = ǫ · h C π ( π ( x ) g ) , C π g i 6 = 0 . Secondly, by choosing ǫ sufficiently small we also have that h C π g, C π g i = k C π g k + ǫ · h C π ( π ( x ) g ) , C π g i 6 = 0 . We can now state the main result of this section regarding the independence of the test vectors H g of the chosen integrable vector g ∈ A \ { } . Theorem 3.5.
Let π : G → U ( H π ) be an integrable representation. Given two integrable vectors g , g ∈ A \ { } the spaces H g and H g coincide with equivalent norms.Proof. Assume first that the two integrable vectors g , g ∈ A \ { } satisfy h C π g , C π g i 6 = 0. Wepick f ∈ H g and want to show that f ∈ H g , that is, we need to check that W g f ∈ L ( G ). Ashort calculation reveals that( W g f ∗ G W g g ) ( x ) = Z G h f, π ( y ) g ih g , π ( y − x ) g i dµ L ( y )= Z G h f, π ( y ) g ih π ( x ) g , π ( y ) g i dµ L ( y )= hW g f, W g ( π ( x ) g ) i L ( G ) = h C π g , C π g iW g f ( x ) . h C π g , C π g i 6 = 0, we can rearrange and integrate so that kW g f k L ( G ) = kW g f ∗ G W g g k L ( G ) |h C π g , C π g i| ≤ kW g f k L ( G ) kW g g k L ( G ) |h C π g , C π g i| . Let us now tackle the case where g , g ∈ A \ { } satisfy h C π g , C π g i = 0. Again, we assumethat f ∈ H g and we want to show that f ∈ H g . We can by Lemma 3.4 pick an integrable vector g ∈ A \ { } such that h C π g, C π g i i 6 = 0 for i = 1 ,
2. Performing similar calculations as previously, weobtain ( W g f ∗ G W g g ) ∗ G W g g = h C π g , C π g iW g f ∗ G W g g = h C π g , C π g ih C π g, C π g iW g f. We have conceptually used g as a stepping stone between g and g . After a rearrangement, we canintegrate and obtain kW g f k L ( G ) = kW g f ∗ G W g g ∗ G W g g k L ( G ) |h C π g , C π g i||h C π g, C π g i|≤ kW g f k L ( G ) kW g g k L ( G ) kW g g k L ( G ) |h C π g , C π g i||h C π g, C π g i| . It clear from the arguments above that the norms on H g and H g are equivalent.Due to the independence of H g of the integrable vector g ∈ A \ { } , we will use the notation H := H g . It follows from Theorem 3.5 that
A ⊂ H since g ∈ A is in H g by definition. For unimodular groups,the following result shows that we do not need to keep track of both H and A . Proposition 3.6.
We have the equality A = H when π : G → U ( H π ) is an integrable representa-tion of a unimodular group G .Proof. We fix f ∈ H and want to show that f ∈ A . The orthogonality relation in (2.10) for x ∈ G shows that h C π g, C π g i H π h f, π ( x ) f i H π = hW g f, W g ( π ( x ) f ) i L ( G ) . Taking the absolute value and using the intertwining property (2.3), we have k C π g k H π |h f, π ( x ) f i H π | ≤ Z G |W g f ( y ) ||W g f ( x − y ) | dµ ( y ) . Notice that k C π g k = 0 since C π is injective. Hence we can divide by k C π g k and integrate withrespect to x , use Fubini’s theorem, and use the right-invariance of the measure µ to obtain kW f f k L ( G ) ≤ k C π g k Z G Z G |W g f ( y ) ||W g f ( x − y ) | dµ ( y ) dµ ( x )= 1 k C π g k Z G |W g f ( y ) | (cid:18)Z G |W g f ( x − y ) | dµ ( x ) (cid:19) dµ ( y )= 1 k C π g k Z G |W g f ( x − ) | dµ ( x ) Z G |W g f ( y ) | dµ ( y ) . G is unimodular, we can use the substitution x x − to obtain kW f f k L ( G ) ≤ k C π g k Z G |W g f ( x ) | dµ ( x ) Z G |W g f ( y ) | dµ ( y ) ≤ k C π g k kW g f k L ( G ) < ∞ . Thus f ∈ A and the claim follows. Example 3.7.
Let us consider the Schr¨odinger representation ρ r of the reduced Heisenberg group H nr . It follows from (2.6) that for any g ∈ L ( R n ) we have W g g ∈ L ( H nr ) precisely whenever V g g ∈ L ( R n ), where V denotes the STFT and W denotes the wavelet transform corresponding to ρ r . Motivated by this observation, we will work with the STFT instead of the wavelet transform.It is straightforward to check that V g g ∈ S ( R n ) ⊂ L ( R n ) whenever g ∈ S ( R n ) is a smoothand rapidly decaying function, for details see [42, Theorem 11.2.5]. Hence ρ r is an integrablerepresentation. We can by Theorem 3.5 and Proposition 3.6 unambiguously define the Feichtingeralgebra M ( R n ) := H = A = (cid:8) f ∈ L ( R n ) : V f f ∈ L ( R n ) (cid:9) . We obtain from Proposition 3.2 that M ( R n ) is a Banach space. The Feichtinger algebra M ( R n )was first introduced in [25] and gained more widespread attention after its appearance in [42]. Werefer the reader to [50] for a detailed and modern exposition on the Feichtinger algebra. In particular,functions in M ( R ) are automatically continuous by [50, Corollary 4.2]. Since there are plenty ofnon-continuous elements in L ( R n ), this gives an example where H = H π . Let π : G → U ( H π ) be an integrable representation and fix an integrable vector g ∈ A \ { } . Inlight of the previous section, we might prematurely define the coorbit space C o p ( G ) for 1 ≤ p ≤ ∞ to be all f ∈ H π such that W g f ∈ L p ( G ). However, this naive definition suffers from the followingproblem: We will obtain C o p ( G ) = H π for every p ≥
2. Only having interesting coorbit spaces inthe range 1 ≤ p ≤ H π is to small to accommodate a full range1 ≤ p ≤ ∞ of interesting spaces. In this section, we will fix this problem by introducing a largerreference space R and ensuring that everything works the way it should. After this is done, we canconfidently define the coorbit spaces properly in Section 3.3. Definition 3.8.
Let π : G → U ( H π ) be an integrable representation. The space of boundedanti-linear functionals on H is denoted by R and called the reservoir space . Remark.
Implicitly, we have chosen an integrable vector g ∈ A \ { } and are considering H g and thespace R g of bounded anti-linear functionals on H g . However, due to Theorem 3.5 we omit g fromthe notation as it is of minor importance. The reservoir space R will seldom consist of functions inany reasonable sense. If we want to understand when two elements φ, ψ ∈ R are equal, we need totest them on all the elements in H . This is the motivation for calling H the space of test vectors . Lemma 3.9.
There are natural continuous embeddings H ֒ −→ H π ֒ −→ R . More precisely, there is a dense subset D ⊂ L ( R n ) of equivalence classes of functions that does not have acontinuous representative. roof. If φ ∈ R and g ∈ H , we denote the dual pairing φ ( g ) by h φ, g i . We can embed H π into R by letting f ∈ H π act on g ∈ H by f ( g ) := h f, g i H π . To see that the inclusion H π ֒ −→ R is continuous we compute for f ∈ H π that k f k R = sup g ∈H \{ } |h f, g i|k g k H ≤ sup g ∈H \{ } k g k H π k g k H ! k f k H π . The claim follows the continuity of the inclusion H ֒ −→ H π in Proposition 3.2.Given an integrable representation π : G → U ( H π ) we can let π act on the reservoir space R through duality. More precisely, for x ∈ G and φ ∈ R we define π ( x ) φ to be the element in R thatacts on g ∈ H by ( π ( x ) φ )( g ) = h π ( x ) φ, g i := h φ, π ( x − ) g i . This gives an isometric action on R since k π ( x ) φ k R = sup g ∈H \{ } |h π ( x ) φ, g i|k g k H = sup g ∈H \{ } |h φ, π ( x − ) g i|k π ( x − ) g k H = k φ k R , where we used that π acts by isometries on H , see Proposition 3.1. We can now extend the wavelettransform to a duality pairing between H and R as follows: Definition 3.10.
Let π : G → U ( H π ) be an integrable representation. For φ ∈ R and g ∈ H wedefine the (extended) wavelet transform to be the function on G given by W g φ ( x ) := h φ, π ( x ) g i = φ ( π ( x ) g ) = ( π ( x − ) φ )( g ) , x ∈ G. Notice that the definition of the extended wavelet transform is well-defined since H is invari-ant under π . Some authors, e.g. [16], change the notation for the extended wavelet transform toemphasize its domain, while other authors [63] do not change the notation. We have opted for thelatter and will strive to make it clear what the wavelet transform acts on. Proposition 3.11.
Let π : G → U ( H π ) be an integrable representation and fix g ∈ H . Then W g ( R ) ⊂ C b ( G ) and we have the intertwining property W g ( π ( x ) φ ) = L x [ W g φ ] , (3.2) for x ∈ G and φ ∈ R .Proof. The map Γ g : x π ( x ) g is clearly a continuous map Γ g : G → H by Proposition 3.1. Hence W g φ = φ ◦ Γ g is continuous since it can be described as the composition of two continuous maps.The boundedness of W g φ follows from the straightforward computation |W g φ ( x ) | = | φ ( π ( x ) g ) | ≤ k φ k R k π ( x ) g k H = k φ k R k g k H , x ∈ G. Finally, the intertwining property is verified by the computation( W g ( π ( x ) φ )) ( y ) = h π ( x ) φ, π ( y ) g i = h φ, π ( x − y ) g i = L x W g φ ( y ) , x, y ∈ G. Remark.
Although the (extended) wavelet transform W g is well-defined for all g ∈ H , we will forthe most part work with the setting where g ∈ A ⊂ H for convenience. Hence we will primarilystate results for W g when g ∈ A , even though they are sometimes valid for g ∈ H as well.24 xample 3.12. We defined in Example 3.7 the Feichtinger algebra M ( R n ) as the set of test vectorscorresponding to the STFT. The reservoir space R in this setting will be denoted by M ∞ ( R n ).Let us do a concrete calculation in the case n = 1: The Dirac Comb distribution δ Z is definedformally as acting on functions f : R → C by δ Z ( f ) := ∞ X n = −∞ f ( n ) . (3.3)The expression (3.3) is obviously not always well defined. It follows from [50, Corollary 5.9] that δ Z ∈ M ∞ ( R ). For g ( t ) := e − t ∈ S ( R ) ⊂ M ( R ) and ( x, ω ) ∈ R we have the explicit computation V g δ Z ( x, ω ) = δ Z ( M − ω T − x g ) = δ Z (cid:16) e − πiωt e − ( t − x ) (cid:17) = ∞ X n = −∞ e − πiωn e − ( n − x ) . An interesting observation is that V g δ Z (0 , ω ) = ϑ ( z, τ ) , where τ = i/π , z = − ω , and ϑ is the Jacobi theta function omnipresent in complex analysis.
Lemma 3.13.
Let π : G → U ( H π ) be an integrable representation and fix g ∈ A \ { } . Then linearcombinations of elements of the form π ( x ) g for x ∈ G constitute a dense subspace of H with respectto the norm on H . Moreover, if g is admissible then we have the reproducing formula W g φ = W g φ ∗ G W g g, for any φ ∈ R .Remark. Originally the density statement in Lemma 3.13 was proved by showing a minimalitystatement regarding the space H . More precisely, it was shown in [28, Corollary 4.8] that H is theminimal π -invariant Banach space inside H π where π acts isometrically and such that A ∩ H = { } . A different proof of the density statement in Lemma 3.13 was given in [63, Lemma 2.4.7] usingBochner integration. The reader can also find a proof of the convolution statement in [63, Lemma2.4.8], again using Bochner integration. We have opted to not present a proof of Lemma 3.13 as itis mostly a technical tool.
Corollary 3.14.
Let π : G → U ( H π ) be an integrable representation and fix an integrable vector g ∈ A \ { } . Then W g : R → L ∞ ( G ) is injective.Proof. Assume that W g φ ( x ) = φ ( π ( x ) g ) = 0 for every x ∈ G . Then Lemma 3.13 shows that φ = 0since the span of the elements π ( x ) g for x ∈ G is a dense subspace of H .Notice that for an integrable vector g ∈ A \ { } we have by definition that W g : H → L ( G ).Hence we can consider the adjoint map W ∗ g : L ∞ ( G ) → R defined by the relation hW ∗ g ( F ) , f i R , H = h F, W g f i L ∞ ( G ) ,L ( G ) = Z G F ( x ) W g f ( x ) dµ L ( x ) = Z G F ( x ) h π ( x ) g, f i dµ L ( x ) , for F ∈ L ∞ ( G ) and f ∈ H . The adjoint map W ∗ g : L ∞ ( G ) → R can hence be written weakly as W ∗ g ( F ) = Z G F ( x ) π ( x ) g dµ L ( x ) , F ∈ L ∞ ( G ) . roposition 3.15. Let π : G → U ( H π ) be an integrable representation and fix an integrable vector g ∈ A \ { } . The adjoint map W ∗ g : L ∞ ( G ) → R satisfies W g (cid:0) W ∗ g ( F ) (cid:1) = F ∗ G W g g, W ∗ g ( W g φ ) = φ, for F ∈ L ∞ ( G ) and φ ∈ R .Proof. For x ∈ G a straightforward computation shows that W g (cid:0) W ∗ g ( F ) (cid:1) ( x ) = hW ∗ g ( F ) , π ( x ) g i R , H = h F, W g ( π ( x ) g ) i L ∞ ( G ) ,L ( G ) = ( F ∗ G W g g )( x ) . (3.4)Finally, we need to show that the map W ∗ g ◦ W g : R → R is in fact the identity map. For φ ∈ R wehave from (3.4) and Lemma 3.13 that W g ( W ∗ g ( W g φ )) = W g φ ∗ G W g g = W g φ. The injectivity of W g : R → L ∞ ( G ) ensured by Corollary 3.14 shows that W ∗ g ( W g φ ) = φ .The following result reveals a deep connection between the extended wavelet transform andconvolutions on the group G . Theorem 3.16.
Let π : G → U ( H π ) be an integrable representation and fix an integrable vector g ∈ A \ { } . A function F ∈ L ∞ ( G ) satisfies the convolution relation F = F ∗ G W g g precisely whenit can be written uniquely as F = W g φ for some φ ∈ R .Proof. If F ∈ L ∞ ( G ) is such that F = F ∗ G W g g , then Proposition 3.15 shows that F = W g ( φ )where φ := W ∗ g ( F ). Moreover, the description F = W g φ is necessarily unique due to the injectivityof W g : R → L ∞ ( G ). Conversely, assume that F ∈ L ∞ ( G ) satisfies F = W g φ for some φ ∈ R .Then we have from Proposition 3.15 that W ∗ g ( F ) = W ∗ g ( W g φ ) = φ. Thus W g ( W ∗ g ( F )) = W g φ = F . The claim follows from a final application of Proposition 3.15. Remark.
We mentioned in Example 3.7 that the space S ( R n ) is included in the Feichtinger algebra M ( R n ). Hence we have by Lemma 3.9 the inclusions S ( R n ) ⊂ M ( R n ) ⊂ L ( R n ) ⊂ M ∞ ( R n ) ⊂ S ′ ( R n ) , where the set of tempered distributions S ′ ( R n ) is the dual space of S ( R n ). We can view the pair (cid:0) M ( R n ) , M ∞ ( R n ) (cid:1) as a refinement of the pair ( S ( R n ) , S ′ ( R n )). A time-frequency analysis enthu-siast might even use the word “improvement” since the Feichtinger algebra M ( R n ) is, in contrastwith S ( R n ), a Banach space. Now that all the pieces are in place we will define the coorbit spaces. These are the main objectsof study for this survey, and we spend a decent amount of time deriving their basic properties.
Definition 3.17.
Let π : G → U ( H π ) be an integrable representation and fix an integrable vector g ∈ A \ { } . The coorbit space C o p ( G ) consists of all elements in the reservoir space φ ∈ R suchthat W g φ decays fast enough to be in L p ( G ). Precisely, we define for each 1 ≤ p ≤ ∞ the space C o p ( G ) := C o πp ( G ) := { φ ∈ R : W g φ ∈ L p ( G ) } , with the norm k φ k C o p ( G ) := kW g φ k L p ( G ) .
26e will only use the full notation C o πp ( G ) in Section 3.8 when we are dealing with multiplerepresentations. The observant reader will have noticed that we did not mention the integrablevector g ∈ A \ { } in the notation C o p ( G ). This is because, as probably suspected, the coorbitspaces C o p ( G ) do not depend on the choice of integrable vector, see [28, Section 5.2] for details. Example 3.18.
Let G be a compact group and let π : G → U ( H π ) be an irreducible representation.Then π is automatically integrable since any g ∈ H π satisfies Z G |W g g ( x ) | dµ L ( x ) ≤ kW g g k L ∞ ( G ) · µ L ( G ) < ∞ . Here we used that the Haar measure of a compact group is finite, see [21, Proposition 1.4.5].Moreover, it is clear that every g ∈ H π satisfies W g g ∈ L p ( G ) for all 1 ≤ p ≤ ∞ . Thus all thecoorbit spaces coincide, that is, C o p ( G ) = H π for all 1 ≤ p ≤ ∞ . Moreover, we mentioned inExample 2.25 that the space H π is necessarily finite-dimensional whenever G is compact. Hencecoorbit spaces are rather dull when considering compact groups.Coorbit spaces associated with a commutative group G are even more boring: In this caseCorollary 2.14 ensures that H π is one-dimensional. From this, it is easy to check that an integrablerepresentation π : G → U ( H π ) can only exist whenever G is compact. Henceforth we will only beinterested in coorbit spaces corresponding to locally compact groups that are both non-compactand non-commutative. Remark.
Before we proceed, it is instructive to consider how the definition of the coorbit spaces canbe generalized. • One could allow p to take values in (0 ,
1) as well. This would make the spaces C o p ( G ) for p ∈ (0 ,
1) quasi-normed spaces instead of normed spaces. We will not consider this extension,and refer the reader to [63] for basic results in this direction. • We can consider weighted coorbit spaces C o p,w ( G ) where w : G → (0 , ∞ ) is a weight function.To do this, one must first incorporate weights into the definition of analyzing vectors A w and test vectors H w . We will briefly go through this extension in Section 3.5. The weightedextension offer mostly technical challenges rather than conceptual ones. As such, we feelcontent with supplying the proofs only in the unweighted setting. We will however providethe reader the proper references whenever we leave out details. • One could go a step further and consider the coorbit space C o ( Y ), where Y is a solid and translation invariant Banach space of functions on G . We omit the precise definitions hereand refer the reader to the original papers [28, 29, 30] as well as Voigtlaender’s Ph.D. thesis[63] for more on the theory in this level of generality. Most concrete applications of coorbittheory use weighted L p -spaces, or mixed-norm L p,q spaces as in the following example. Example 3.19.
Let us again consider the STFT. In this case, we have the notation H = M ( R n )and R = M ∞ ( R n ). The coorbit spaces in this setting are called the modulation spaces . Moreexplicitly, for a non-zero g ∈ M ( R n ) and 1 ≤ p ≤ ∞ the space M p ( R n ) consists of elements f ∈ M ∞ ( R n ) such that k f k M p ( R n ) := (cid:18)Z R n | V g f ( x, ω ) | p dx dω (cid:19) p < ∞ . It will be clear from Proposition 3.22 that M ( R n ) = L ( R n ).27e can generalize the modulation spaces slightly by using mixed-norm L p,q spaces. More pre-cisely, we define the mixed-norm modulation spaces M p,q ( R n ) for 1 ≤ p, q ≤ ∞ as the elements f ∈ M ∞ ( R n ) such that k f k M p,q ( R n ) := Z R n (cid:18)Z R n | V g f ( x, ω ) | p dx (cid:19) qp dω ! q < ∞ . Notice that M p,p ( R n ) = M p ( R n ). This extension allows us to consider different levels of integra-bility in time and frequency. We remark that the space M ∞ , ( R n ) has appeared in the theory ofpseudodifferential operators under the name Sj¨ostrand’s class . We refer the reader to [43] for moreon Sj¨ostrand’s class in the context of time-frequency analysis. More general information regardingthe mixed modulation spaces M p,q ( R n ) can be found in [42, Chapter 11 and 12].Most of the basic properties of coorbit spaces will be derived in Section 3.4. Before this, we willestablish a powerful result known as the correspondence principle . In essence, the correspondenceprinciple states that one can identify the abstract coorbit space C o p ( G ) with the space M p ( G ) := { F ∈ L p ( G ) : F = F ∗ G W g g } . Notice that M p ( G ) is more concrete that C o p ( G ), in the sense that it consists of functions onthe group G in question. The fact that the wavelet transform W g for g ∈ A \ { } provides theisomorphism between C o p ( G ) and M p ( G ) makes the result even more conceptually pleasing. Theorem 3.20 (Correspondence Principle) . Let π : G → U ( H π ) be an integrable representationand fix an integrable vector g ∈ A \ { } . Then for every ≤ p ≤ ∞ the wavelet transform W g is anisomorphism W g : C o p ( G ) ∼ −→ M p ( G ) . Proof.
It follows immediately from Theorem 3.16 that W g ( C o p ( G )) ⊂ M p ( G ). Hence it only remainsto show that any F ∈ M p ( G ) is in fact of the form F = W g f for some f ∈ C o p ( G ). Notice that W g g ∈ L q ( G ) for all 1 ≤ q ≤ ∞ since W g g ∈ L ( G ) ∩ L ∞ ( G ) ⊂ L q ( G ) . We choose q such that p − + q − = 1, with the obvious caveats for p = 1 , ∞ . Then F = F ∗ W g g ∈ L ∞ ( G ) . Hence the machinery in Theorem 3.16 implies that F = W g f for some f ∈ R . We have that f ∈ C o p ( G ) by definition since F ∈ L p ( G ). In this section we derive the basic properties of coorbit spaces. The reader should pay specialattention to how the correspondence principle we proved in Theorem 3.20 is utilized in several ofthe proofs in this section.
Theorem 3.21.
Let π : G → U ( H π ) be an integrable representation. Then the coorbit spaces C o p ( G ) are π -invariant Banach spaces on which π acts by isometries. roof. We fix an integrable vector g ∈ A \ { } . Let us first show that k · k C o p ( G ) is in fact a norm.The only non-trivial point is the positive-definiteness. Assume that kW g f k L p ( G ) = 0 for some f ∈ C o p ( G ). Then W g f is zero almost everywhere as a function on G . Since W g f is a continuousfunction on G by Proposition 3.11, we have that W g f is identically zero. Since W g : R → L ∞ ( G )is injective, we conclude that f = 0.To show completeness, we assume that { f n } n ∈ N is a Cauchy sequence in C o p ( G ). Then {W g f n } n ∈ N is a Cauchy sequence in L p ( G ). By completeness of L p ( G ), there exists F ∈ L p ( G ) such that W g f n → F in L p ( G ). It follows that F ∗ W g g = (cid:16) lim n →∞ W g f n (cid:17) ∗ G W g g = lim n →∞ ( W g f n ∗ G W g g ) = lim n →∞ W g f n = F. We can now use the correspondence principle in Theorem 3.20 to conclude that F = W g f for some f ∈ C o p ( G ). Hence the coorbit spaces C o p ( G ) are complete since k f n − f k C o p ( G ) = kW g f n − W g f k L p ( G ) → . Finally, if f ∈ C o p ( G ) and x ∈ G then we use (3.2) to obtain k π ( x ) f k C o p ( G ) = kW g ( π ( x ) f ) k L p ( G ) = k L x W g f k L p ( G ) = kW g f k L p ( G ) = k f k C o p ( G ) . The following proposition shows that the spaces H , H π , and R all have descriptions in termsof coorbit spaces. Proposition 3.22.
Let π : G → U ( H π ) be an integrable representation. We have the descriptions C o ( G ) = H , C o ( G ) = H π , C o ∞ ( G ) = R . Proof.
As usual, we fix an integrable vector g ∈ A \ { } . The statement C o ∞ ( G ) = R is clear fromthe definition of C o ∞ ( G ) since every φ ∈ R satisfies W g φ ∈ L ∞ ( G ) by Proposition 3.11. We havethat H ⊂ C o ( G ) and H π ⊂ C o ( G ) through the inclusions in Lemma 3.9. Conversely, assume that f ∈ C o ( G ). Then W g f ∈ L ( G ) and satisfies by the correspondence principle in Theorem 3.20 theconvolution relation W g f = W g f ∗ G W g g. However, in Theorem 2.32 we showed that F F ∗ G W g g is the projection from L ( G ) to thespace W g ( H π ). Hence we conclude that W g f = W g h for some h ∈ H π . Since W g : R → L ∞ ( G ) isinjective we have that f = h as elements in R . Moreover, the injectivity of the inclusion H π ֒ −→ R forces f ∈ H π , and thus the claim C o ( G ) = H π follows. Since L ( G ) ∩ L ∞ ( G ) ⊂ L ( G ), we canrepeat the same argument for f ∈ C o ( G ) and find that f ∈ H π . As H is by definition the set ofelements f ∈ H π such that W g f ∈ L ( G ), we have that C o ( G ) = H . Remark.
The proof of Proposition 3.22 shows that C o p ( G ) ⊂ H π for all p ∈ [1 ,
2] since then L p ( G ) ∩ L ∞ ( G ) ⊂ L ( G ).The following result shows that the coorbit spaces C o p ( G ) inherit their duality properties fromthe L p ( G )-spaces. For a proof of this result, we refer the reader to [29, Theorem 4.9]. Proposition 3.23.
Let π : G → U ( H π ) be an integrable representation. The coorbit spaces C o p ( G ) for ≤ p < ∞ satisfy the duality C o p ( G ) ′ = C o q ( G ) , p + 1 q = 1 . In particular, the coorbit spaces C o p ( G ) are reflexive Banach spaces for < p < ∞ . xample 3.24. Let us again consider the affine group Aff together with the wavelet representation π : Aff → U ( L ( R )) given by π ( b, a ) f ( x ) := T b D a f ( x ) = 1 p | a | f (cid:18) x − ba (cid:19) . We showed in Example 2.27 that π is a square integrable representation. A straightforward com-putation shows that π is in fact integrable by considering a non-zero function g ∈ S ( R ) such that F ( g ) is supported on [ r, s ] for r, s ∈ (0 , ∞ ). Hence for any integrable vector g ∈ A \ { } we obtainfor each 1 ≤ p < ∞ the affine coorbit space C o p (Aff) defined by C o p (Aff) := ( f ∈ R : k f k C o p (Aff) := (cid:18)Z Aff |W g f ( b, a ) | p db daa (cid:19) p < ∞ ) . As usual, the case p = ∞ is defined with the supremum. We immediately get from Theorem 3.21that C o p (Aff) is a Banach space for each 1 ≤ p ≤ ∞ on which the wavelet representation π acts byisometries. In this section, we will discuss how coorbit spaces can be generalized to include weights. This isusually done right from the beginning in the literature, see e.g. [28, 29, 30, 16, 63]. However, wehave opted to introduce this separately so that coorbit spaces could initially be introduced withminimal technicalities. As weights do not introduce anything conceptually new, this section mostlyconsists of technicalities that invoke feelings of d´ej`a vu.
Definition 3.25.
Let G be a locally compact group. Given any continuous function w : G → (0 , ∞ )we can form the weighted L p -space L pw ( G ) for 1 ≤ p ≤ ∞ consisting of all equivalence classes ofmeasurable function f : G → C such that k f k L pw ( G ) := k f · w k L p ( G ) < ∞ . We say that a continuous function w : G → (0 , ∞ ) is a weight function if it is sub-multiplicative ,that is, w satisfies w ( xy ) ≤ w ( x ) w ( y ) for all x, y ∈ G . Remark.
The reader should be aware that the conditions that goes into the term weight function (or simply weight ) differs quite a bit from author to author: In [63] a sub-multiplicative weightis not assumed to be continuous, only measurable. It turns out that a not necessarily continuoussub-multiplicative weight is automatically bounded on compact sets by [63, Theorem 2.2.22]. Aweight w in [16, Chapter 3] is assumed to be symmetric , meaning that w ( x ) = w ( x − ) for all x ∈ G .The symmetry assumption automatically gives that w ≥
1. If w is a not necessarily symmetricweight function on G such that w ≥
1, then L pw ( G ) ֒ −→ L p ( G ) is a continuous embedding since k f k L p ( G ) = (cid:18)Z G | f ( x ) | p dµ L ( x ) (cid:19) p ≤ (cid:18)Z G | f ( x ) | p w ( x ) p dµ L ( x ) (cid:19) p = k f k L pw ( G ) , for all f ∈ L pw ( G ). Example 3.26.
Consider the function w on G = (0 , ∞ ) given by w ( x ) := e | log( x ) | = ( x if x ≥ x , if x < .
30t is straightforward to verify that w is a symmetric weight function. The condition for a measurablefunction f : G → C to be in L w ( G ) takes the form Z | f ( x ) | x dx + Z ∞ | f ( x ) | dx < ∞ . Definition 3.27.
Let π : G → U ( H π ) be an irreducible unitary representation of the locally compactgroup G and fix a weight function w : G → (0 , ∞ ). The representation π is called w -integrable ifthere exists a non-zero element g ∈ H π such that W g g ∈ L w ( G ). We use the notation A w := (cid:8) g ∈ H π : W g g ∈ L w ( G ) (cid:9) . Similarly as before, we fix g ∈ A w \ { } and define the space of w -test vectors H w,g as the elements f ∈ H π such that W g f ∈ L w ( G ).The proof of the following result illustrates the usefulness of the sub-multiplicative condition. Lemma 3.28.
Let π : G → U ( H π ) be a w -integrable representation and fix g ∈ A w \ { } . Then π acts continuously and invariantly on H w,g .Proof. We fix f ∈ H w,g and compute for x ∈ G that k π ( x ) f k H w,g = Z G |W g ( π ( x ) f )( y ) | w ( y ) dµ L ( y )= Z G |W g ( f )( x − y ) | w ( y ) dµ L ( y )= Z G |W g ( f )( y ) | w ( xy ) dµ L ( y ) . By using the sub-multiplicative condition we end up with k π ( x ) f k H w,g ≤ w ( x ) Z G |W g ( f )( y ) | w ( y ) dµ L ( y ) = w ( x ) · k f k H w,g . We can now use Lemma 3.28 to see that the space H w,g is dense in H π for all g ∈ A w \ { } .It is straightforward to check that the space L w ( G ) is invariant under both the left-translationoperator and the right-translation operator. This fact is sufficient for Lemma 3.4 to go through inthe weighted setting. Finally, only minor changes are needed in Proposition 3.2, Theorem 3.5, andProposition 3.6 to obtain the weighted statements. Hence H w,g does not depend on the choice of g ∈ A w \ { } and we simply write H w := H w,g . Example 3.29.
A class of commonly used symmetric weight functions on R n is given by v s ( x, ω ) := (1 + | x | + | ω | ) s , ( x, ω ) ∈ R n , s ≥ . The family v s is sometimes referred to as the polynomial weights . For the STFT we can use thepolynomial weights to define the weighted Feichtinger algebra M s ( R n ) := H v s . The inequality v s ≤ v t , ≤ s ≤ t implies the inclusion M t ( R n ) ⊂ M s ( R n ). In particular, we have M s ( R n ) ⊂ M ( R n ) for all s ≥ . It is straightforward to check that M s ( R n ) still contains the rapidly decaying and smooth functions S ( R n ) for all s ≥
0. Is there anything more than S ( R n ) contained in all of the weighted Feichtingeralgebras M s ( R n ) for s ≥
0? By [42, Proposition 11.3.1] the answer is negative and we can write S ( R n ) = \ s ≥ M s ( R n ) . efinition 3.30. Let π : G → U ( H π ) be a w -integrable representation. We define the w -reservoirspace R /w as the space of bounded anti-linear functionals on H w .The duality between H w and R /w is again denoted by φ ( g ) = h φ, g i for g ∈ H w and φ ∈ R /w .Lemma 3.9 goes through directly with the new notational changes and we have the inclusions H w ֒ −→ H π ֒ −→ R /w . The action of π on R /w is defined in the same way as in Section 3.2. We can again define the (extended) wavelet transform by the formula W g φ ( x ) := h φ, π ( x ) g i , g ∈ H w , φ ∈ R /w . Remark.
The reader should be aware that 1 /w is not in general a weight function, even when w : G → (0 , ∞ ) is a symmetric weight function. However, the failure of 1 /w to be sub-multiplicativecan be remedied: If w is symmetric, then we can write for x, y ∈ G that w ( x ) = w ( xyy − ) ≤ w ( xy ) w ( y − ) = w ( xy ) w ( y ) . Hence 1 w ( xy ) ≤ w ( x ) w ( y ) . This relation suffices in most settings.The proof of Lemma 3.13 in [63, Lemma 2.4.7 and Lemma 2.4.8] is stated in the weightedcase. Finally, Corollary 3.14, Proposition 3.15, and Theorem 3.16 are almost verbatim the same aspreviously. The only thing worth remarking is that the space of bounded anti-linear functionals on L w ( G ) is L ∞ /w ( G ). This motivates the notation R /w . Example 3.31.
For the STFT we use the notation M ∞ /s ( R n ) := R /v s , where v s for s ≥ S ( R n )has the dual version S ′ ( R n ) = [ s ≥ M ∞ /s ( R n ) . Hence the pair ( S ( R n ) , S ′ ( R n )) works as limiting cases for respectively the weighted Feichtingeralgebras M s ( R n ) and the weighted reservoir spaces M ∞ /s ( R n ) for s ≥ Definition 3.32.
Let π : G → U ( H π ) be a w -integrable representation and fix a w -integrable vector g ∈ A w \ { } . The (weighted) coorbit space C o p,w ( G ) for 1 ≤ p ≤ ∞ is given by the straightforwardextension C o p,w ( G ) := { φ ∈ R /w : W g φ ∈ L pw ( G ) } , with the norm k φ k C o p,w ( G ) := kW g φ k L pw ( G ) . As previously, the coorbit spaces C o p,w ( G ) do not depend on the choice of w -integrable vector g ∈ A w \ { } , see [28, Section 5.2] for details. It is clear that C o ∞ , /w ( G ) = R /w . xample 3.33. We define the weighted modulation spaces M p,qs ( R n ) for 1 ≤ p, q ≤ ∞ and s ≥ f ∈ M ∞ /s ( R n ) such that k f k M p,qs ( R n ) := Z R n (cid:18)Z R n | V g f ( x, ω ) | p (1 + | x | + | ω | ) ps dx (cid:19) qp dω ! q < ∞ , where g ∈ A v s \ { } is fixed. Since the reduced Heisenberg group H nr is unimodular, it follows fromthe weighted version of Proposition 3.6 that A v s = H v s = M s ( R n ) . It is common in practice to choose g ∈ S ( R n ) \ { } , which is valid since S ( R n ) ⊂ M s ( R n ) for all s ≥
0. Moreover, one can choose the reservoir to be S ′ ( R n ) instead of M ∞ /s ( R n ) without changingthe weighted modulation spaces. It is possible to use different weights to obtain other weightedmodulation spaces. We refer the reader to [42, Section 11.4] to see how one can define weightedmodulation spaces where the weights have exponential growth.The completeness of L pw ( G ) for a weight function w : G → (0 , ∞ ) allows us to extend the firststatement in Theorem 3.21 to the weighted setting. The second statement in Theorem 3.21 has tobe altered to say that π acts continuously on the weighted coorbit spaces C o p,w ( G ); this uses thesame argument we gave in the proof of Lemma 3.28. The statement in Proposition 3.22 is valid inthe weighted setting with the nessesary changes. More precisely, for g ∈ A w \{ } we let H π,w denotethe elements f ∈ H π such that W g f ∈ L w ( G ). Then we can adapt the proof of Proposition 3.22 tosee that C o ,w ( G ) = H w , C o ,w ( G ) = H π,w , C o ∞ , /w ( G ) = R /w . Finally, the duality statement in Proposition 3.23 is still valid in the weighted setting with thenessesary changes, see [29, Theorem 4.9] for details. Before moving on, we summarize the mostimportant results regarding the weighted coorbit spaces in one theorem so that we have precisestatements we can reference later in the survey.
Theorem 3.34.
Let π : G → U ( H π ) be a w -integrable representation where w : G → (0 , ∞ ) isa weight function. Fix a w -integrable vector g ∈ A w \ { } . Then the coorbit spaces C o p,w ( G ) for ≤ p ≤ ∞ satisfy the following properties:a) The coorbit spaces C o p,w ( G ) are Banach spaces on which the representation π acts invariantlyand continuously.b) An element F ∈ L pw ( G ) satisfies the convolution relation F = F ∗ G W g g if and only if F = W g f for some f ∈ C o p,w ( G ) .c) We have the identifications C o ,w ( G ) = H w , C o ,w ( G ) = H π,w , C o ∞ , /w ( G ) = R /w . d) The coorbit spaces C o p,w ( G ) for ≤ p < ∞ satisfy the duality relation C o p,w ( G ) ′ = C o q, /w ( G ) , p + 1 q = 1 , where C o p, /w ( G ) := { φ ∈ R /w : W g φ ∈ L p /w ( G ) } . xample 3.35. Consider the function w s : Aff → (0 , ∞ ) for s ≥ w s ( b, a ) := | a | − s . The computation w s ( b, a ) w s ( d, c ) = | a | − s | c | − s = | ac | − s = w s (( b, a ) · ( d, c )) , ( b, a ) ( d, c ) ∈ Aff , shows that w s is multiplicative, and hence clearly a weight function. The argument in Example 3.24can be extended to show that the wavelet representation π : Aff → U ( L ( R )) is w s -integrable forany s ≥
0. Thus we can consider the weighted affine coorbit spaces C o p,w s (Aff). It turns out that C o p,w s (Aff) = ˙ B s − + p p ( R ) , where ˙ B sp ( R ) denotes the homogeneous Besov space in classical harmonic analysis with smoothnessparameter s ∈ R and integrability parameter 1 ≤ p ≤ ∞ . We refer the reader to [28] for details ofthis fascinating connection. We have so far introduced the coorbit spaces and derived their basic properties. The message thatshould be drawn from Theorem 3.34 is that coorbit spaces form a well-behaved class of Banachspaces. Nevertheless, the reader might find herself wondering what the fuzz is all about. Construct-ing function spaces is commonplace in modern mathematics, so it is maybe unclear why coorbitspaces offer something special. The goal of this section is to convince the reader that the coorbitspaces are deeply connected with the geometry of the underlying locally compact group. Moreover,this connection is inherently practical as it furnishes us with a natural way to discretize elementsin coorbit spaces as we mentioned in (1.1). This makes coorbit spaces novel because they form abridge between geometry, representation theory, and approximation theory.Let us start by precisely stating the continuous reconstruction formula for coorbit spaces. Fixa weight function w : G → (0 , ∞ ) and a w -integrable representation π : G → U ( H π ). Then for f ∈ C o p,w ( G ) we can use the weighted version of Proposition 3.15 to write f = W ∗ g ( W g f ) = Z G W g f ( x ) π ( x ) g dµ L ( x ) , (3.5)for g ∈ A w \ { } . We refer to (3.5) as the continuous reconstruction formula for C o p,w ( G ).What does a discretization of (3.5) look like? Replacing the integral with summation, we hopeto express f ∈ C o p,w ( G ) as the discrete superposition f = X i ∈ I c i ( f ) π ( x i ) g, (3.6)where ( c i ( f )) i ∈ I are coefficients that depend on f and { x i } i ∈ I ⊂ G is a chosen countable collectionof points. We note that (3.6) should be interpreted as convergence in the norm on C o p,w ( G ) for1 ≤ p < ∞ . When p = ∞ we interpret (3.6) as convergence in the weak ∗ -topology. In theliterature, expansions on the form (3.6) are sometimes called atomic decompositions as the element g is considered an atom from which all other relevant functions are constructed. Three naturalquestions emerge: • How can we chose the collection { x i } i ∈ I ⊂ G such that (3.6) converges appropriately? • How does the size of f ∈ C o p,w ( G ) affect the size of ( c i ( f )) i ∈ I in a suitable norm?34 Is it possible to choose the coefficients ( c i ( f )) i ∈ I to depend linearly on f ?Before we answer the questions above in Theorem 3.40 we will borrow some terminology from largescale geometry. This will provide a conceptual language for discussing discretizations. Definition 3.36.
Let X be a non-empty set. We will refer to a collection of non-empty subsets Q = ( Q i ) i ∈ I as an admissible covering for X if X = ∪ i ∈ I Q i andsup i ∈ I (cid:12)(cid:12)(cid:12)n j ∈ I (cid:12)(cid:12)(cid:12) Q i ∩ Q j = ∅ o(cid:12)(cid:12)(cid:12) < ∞ . (3.7)Intuitively, the condition (3.7) states that each Q i ∈ Q can not have to many neighbors. Givenan admissible covering Q = ( Q i ) i ∈ I for a non-empty set X , we call a sequence Q i , . . . , Q i k ∈ Q with x ∈ Q i and y ∈ Q i k a Q - chain from x to y of length k whenever Q i l ∩ Q i l +1 = ∅ for every1 ≤ l ≤ k −
1. The notation Q ( k, x, y ) will be used to denote all Q -chains of length k from x to y . An admissible covering Q on a set X will be called a concatenation if for every pair of points x, y ∈ X there exists a positive number k ∈ N such that Q ( k, x, y ) = ∅ . The idea, originating from[27], is to consider a metric d Q that incorporates closeness relative to the covering Q . This idea hasmore recently been further investigated in [4, 53]. Formally, we have the following definition. Definition 3.37.
Consider a concatenation Q = ( Q i ) i ∈ I for a non-empty set X . Define the metric d Q on X by the rule d Q ( x, x ) = 0 for all x ∈ X and d Q ( x, y ) = inf { k : Q ( k, x, y ) = ∅} , x, y ∈ X, x = y. It is straightforward to check that d Q is indeed a metric on X . Notice that d Q ( x, y ) < ∞ forall x, y ∈ X precisely because we assume that Q is a concatenation. We will refer to ( X, d Q ) as the associated metric space to the concatenation Q . A subset N ⊂ X is called a net if there exists afixed constant C > x ∈ X there is y ∈ N such that d Q ( x, y ) < C . Definition 3.38.
Let (
X, d X ) and ( Y, d Y ) be two metric spaces. We say that a map f : X → Y isa quasi-isometry if f ( X ) is a net in ( Y, d Y ) and there exist fixed constants C, L >
L d X ( x, y ) − C ≤ d Y ( f ( x ) , f ( y )) ≤ Ld X ( x, y ) + C, for every x, y ∈ X . Remark.
Notice that a quasi-isometry f : X → Y is a generalization of an isometry where themap f does not need to be injective nor surjective. This is a suitable notion for comparing metricspaces of different cardinalities. As an example, the inclusion i : Z ֒ → R is a quasi-isometry whenconsidering the standard metrics.Let us now focus on the setting we are interested in. Given a locally compact group G wefix a compact set Q with non-empty interior that contains the identity element e ∈ G . Then thecollection Q cont := ( x · Q ) x ∈ G is a cover for G that is typically not admissible. However, it is always possible to find a subfamily N = { x i } i ∈ I ⊂ G such that Q := ( x i · Q ) i ∈ I is admissible by [26, Theorem 4.1 (A)]. This way ofobtaining N is non-constructive and one usually relies on an understanding of the geometry of G in practical situations to construct N . We refer to Q as the uniform covering corresponding to G with reference set Q . When G is path-connected the covering Q is actually a concatenation, see354, Lemma 3.1]. Hence we obtain an associated metric d Q on G . Maybe surprisingly, the resultingmetric space ( G, d Q ) does not depend (up to quasi-isometry) on the choice of N by [26, Theorem4.1 (B)]. In light of this, we refer to the metric d Q as the uniform metric and the space ( G, d Q ) asthe uniform metric space corresponding to a path-connected locally compact group G . Althoughthe metric d Q is left-invariant, it is almost never compatible with the underlying topology of G . Example 3.39.
Consider the group G = R with the reference set Q = [ − , Q cont = ( x + Q ) x ∈ R = ([ x − , x + 1]) x ∈ R . The subfamily N = Z makes Q := ( n + Q ) n ∈ Z = ([ n − , n + 1]) n ∈ Z into a concatenation. Due to the left invariance of the metric d Q , it is completely determined by d Q (0 , x ) = ⌈ x ⌉ , x > , where ⌈ x ⌉ denotes the ceiling function of x ∈ R . Remark.
The points { x i } i ∈ I such that Q = ( x i · Q ) i ∈ I is the uniform covering of G are only candidates for points where the atomic discretization (3.6) is valid. As an extreme example, consider when G is compact and we pick the reference set Q = G . Then Q = { e · Q } = { Q } is the uniform covering.However, one does not generally have a discretization f = c ( f ) · π ( e ) g = c ( f ) · g, for all f ∈ C o p ( G ) since C o p ( G ) is not necessarily one-dimensional. The problem here is that thereference set Q is to large.The following theorem is the main result regarding atomic decompositions. Theorem 3.40 (Atomic Decomposition Theorem) . Let π : G → U ( H π ) be a w -integrable represen-tation, where w : G → (0 , ∞ ) is a weight function. For well-behaved g ∈ A w \ { } we have for anysufficiently small reference set Q and any ≤ p ≤ ∞ the following properties: • For f ∈ C o p,w ( G ) we have the discrete reconstruction formula f = X i ∈ I c i ( f ) π ( x i ) g, where Q = ( x i · Q ) i ∈ I is the uniform covering corresponding to the reference set Q . Thesequence ( c i ( f )) i ∈ I depends linearly on f . Moreover, there exists a constant C A > notdepending on f such that k ( c i ( f )) i ∈ I k l pw ( I ) ≤ C A k f k C o p,w ( G ) . • Given a sequence ( c i ) i ∈ I ∈ l pw ( I ) we can construct f = X i ∈ I c i π ( x i ) g such that k f k C o p,w ( G ) ≤ C R k ( c i ) i ∈ I k l pw ( I ) , where C R > is a constant not depending on ( c i ) i ∈ I . emark. There are a few details regarding Theorem 3.40 that should be clarified: • For a discrete index set I and a function w : I → (0 , ∞ ), the space l pw ( I ) for 1 ≤ p < ∞ denotes the sequences ( a i ) i ∈ I such that X i ∈ I | a i | p w ( i ) p ! p < ∞ . (3.8)The case p = ∞ is given by replacing summation with supremum. It is straightforward tocheck that l pw ( I ) are Banach spaces with the norm (3.8). In the setting of Theorem 3.40 thefunction w : I → (0 , ∞ ) is obtained by w ( i ) := w ( x i ), where w ( x i ) is the weight function w : G → (0 , ∞ ) evaluated at the point x i ∈ G . We use the same notation for the weightfunction w : G → (0 , ∞ ) and the induced map w : I → (0 , ∞ ) on the index set I . • The requirement that g ∈ A w \{ } should be well-behaved is a technical condition. A sufficientcriterion in general is that W g g belongs to certain Wiener amalgam spaces [16, Theorem 3.15].We refer the reader to [24] and the survey [47] for more details on Wiener amalgam spaces. • The idea for the proof of Theorem 3.40 is to approximate the convolution operator F F ∗ G W g g with special operators involving the wavelet transform. As these ideas are further elaboratedon in [11, Proof of Theorem 24.2.4], we will not go more into this. The full proof of Theorem3.40 can be found in [29, Theorem 6.1] and in [16, Theorem 3.15].Let us use the language of large scale geometry to make Theorem 3.40 more conceptual: Pick asufficiently small reference set Q ⊂ G and the associated family N = ( x i ) i ∈ I corresponding to theuniform covering Q = ( x i · Q ) i ∈ I . Define the trivial map j : N → l pw ( I ) given by j ( x i ) = δ i . Fix awell-behaved element g ∈ A w \ { } and define h : G → C o p,w ( G ) by h ( x ) = π ( x ) g . Finally, we havea reconstruction map R : l pw ( I ) → C o p,w ( G ) given by R (( c i ) i ∈ I ) = X i ∈ I c i π ( x i ) g. Together, these maps form the commutative diagram G C o p,w ( G ) N l pw ( I ) hj R (3.9)When G is equipped with the uniform metric d Q the inclusion N ֒ −→ G in (3.9) is a quasi-isometry.The gist of Theorem 3.40 is that this quasi-isometry carries over to the reconstruction map R in(3.9), where it manifests itself as a norm-equivalence. Looking back at Theorem 3.40, we see that it characterizes the elements in C o p,w ( G ) in terms ofdiscrete expansions. However, if we are given f ∈ R /w , then it might not be obvious to checkwhether f ∈ C o p,w ( G ) with a set of discrete conditions. This leads us to the following question:37 : Given the elements π ( x i ) g for i ∈ I in Theorem 3.40, is it possible to determineif f ∈ C o p,w ( G ) based on the interaction between f and π ( x i ) g for all i ∈ I ?We show in this section that the answer to the question is affirmative. Before stating the result, webriefly discuss Banach frames to put the result into context. Definition 3.41.
Let B be a separable Banach space. Consider a countable subset E = { g i } i ∈ I of continuous anti-linear functionals on B together with an associated sequence space B E on theindex set I . We say that the pair ( E , B E ) is a Banach frame for B if the following two propertiesare satisfied: • The coefficient operator C E : B → B E defined by C E ( f ) = ( h g i , f i ) i ∈ I for f ∈ B satisfies thenorm-equivalence k f k B ≍ kC E ( f ) k B E . (3.10) • There exists a bounded linear map R E : B E → B called the reconstruction operator that is aleft inverse for C E .Explicitly, a reconstruction operator R E : B E → B for the Banach frame ( E , B E ) satisfies R E (( h g i , f i ) i ∈ I ) = f, f ∈ B. The notion of a Banach frame was first considered in [41]. In [8, Proposition 2.4] it was shownthat there exists a Banach frame for any separable Banach space. However, the mere existence ofa Banach frame is not necessarily useful as it might be difficult to both understand and compute.
Example 3.42.
The most well-studied example of a Banach frame is in the case where H = B is a separable Hilbert space. Then, by identifying H with its anti-dual space, we can consider thesequence E = { g i } i ∈ I ⊂ H . Moreover, in this case there is a natural sequence space available,namely l ( I ). Hence the norm equivalence in (3.10) requires that there exists A, B > A k f k H ≤ X i ∈ I |h f, g i i| ≤ B k f k H . It turns out the existence of the reconstruction operator R E is automatically satisfied in this caseby [11, Theorem 3.2.3] and is given by R E (( c i ) i ∈ I ) = X i ∈ I c i g i , ( c i ) i ∈ I ∈ l ( I ) . In fact, the reconstruction operator R E is in this case simply the Hilbert space adjoint of thecoefficient operator! In light of these simplifications, it makes sense to simply refer to the collection E as a frame for the Hilbert space H . Frame theory has a prominent place in modern appliedharmonic analysis, and we refer the reader to [11] for more on this fascinating topic.The following result answers the question posed in the introduction of this section, and we referthe reader to the original paper [41] for a proof. Theorem 3.43.
Consider a w -integrable representation π : G → U ( H π ) where w : G → (0 , ∞ ) isa weight function. Choose a well-behaved g ∈ A w \ { } and any sufficiently small reference set Q .Then the pair E = { π ( x i ) g } i ∈ I , B E = l pw ( I ) is a Banach frame for the coorbit space C o p,w ( G ) , where ≤ p ≤ ∞ and Q = ( x i · Q ) i ∈ I is theuniform covering corresponding to the reference set Q . emark. • Notice that, under the assumptions in Theorem 3.43, the elements in E = { π ( x i ) g } i ∈ I belongto H w . Hence it makes sense for f ∈ C o p,w ( G ) ⊂ R /w to consider the duality pairing W g f ( x i ) = h f, π ( x i ) g i R /w , H w . As such, the coefficient operator C E in this case is simply given by sampling on the points { x i } i ∈ I ⊂ G , that is, C E ( f ) = ( W g f ( x i )) i ∈ I . • The reader should be aware that although the collection E = { π ( x i ) g } i ∈ I is fixed for each1 ≤ p ≤ ∞ , the sequence space B E = l pw ( I ) does indeed depend on p . Since we have defineda Banach frame as the pair ( E , B E ), we are being slightly imprecise when stating that Theo-rem 3.43 provides a single Banach frame for all the coorbit spaces C o p,w ( G ) for 1 ≤ p ≤ ∞ . Example 3.44.
Consider for α, β > Q α,β of R n given by Q α,β := (( αk, βl ) + Q ) k,l ∈ Z n , Q := [ − α, α ] n × [ − β, β ] n . Then for g ∈ S ( R n ) and sufficiently small α, β we have that E = { M βl T αk g } k,l ∈ Z n , B E = l pv s ( Z n ) , is a Banach frame for the modulation space M ps ( R n ) := M p,pv s ( R n ) , where v s for s ≥ E is often called a Gabor system in theliterature. Hence we have the norm-equivalence k f k M ps ( R n ) ≍ X k,l ∈ Z n | V g f ( αk, βl ) | p (1 + | αk | + | βl | ) sp p . The
Schwartz kernel theorem is one of the most influential results in distribution theory. It statesthat any continuous linear operator A : S ( R n ) → S ′ ( R n ) can be represented by a unique distribu-tional kernel K ∈ S ′ ( R n ) in the sense that h Af, g i = h K, f ⊗ g i , f, g ∈ S ( R n ) . (3.11)If K is a locally integrable function, then we have that K is indeed an integral kernel in the sensethat h Af, g i = Z R n K ( x, y ) f ( y ) g ( x ) dy dx, f, g ∈ S ( R n ) . In [5], Hans Georg Feichtinger showed that a kernel theorem is also valid for the modulationspaces. More precisely, he showed that any continuous linear operator A : M ( R n ) → M ∞ ( R n ) canbe represented as in (3.11) with K ∈ M ∞ ( R n ). This result shows that kernel theorems are alsopossible in the Banach space setting. Building on this, the authors in [1] have recently extendedFeichtinger’s kernel theorem to coorbit spaces. In this section, we showcase their results withoutweights for simplicity and refer the reader to the well-written paper [1] for more information.Consider two integrable representations π : G → U ( H ) and π : G → U ( H ). From this weobtain the corresponding coorbit spaces C o π p ( G ) and C o π q ( G ) for all 1 ≤ p, q ≤ ∞ . The goal isto represent any continuous and linear operator A : C o π ( G ) → C o π ∞ ( G ) though a distributionalkernel K in an appropriate sense. The first step is to identify which space the distributional K should be taken from. To do this, we briefly review tensor products of representations.39 efinition 3.45. We can consider the tensor product representation π ⊗ π from G × G tounitary operators on the tensor product H ⊗ H given on simple tensors ψ ⊗ ψ ∈ H ⊗ H by( π ⊗ π )( g , g )( ψ ⊗ ψ ) := π ( g ) ψ ⊗ π ( g ) ψ , ( g , g ) ∈ G × G . It is straightforward to verify that if ψ ∈ H and ψ ∈ H are integrable vectors for respectively π and π , then ψ ⊗ ψ ∈ H ⊗ H is an integrable vector for the tensor product representation π ⊗ π . As such, it makes sense to consider the coorbit space C o p ( G × G ) := C o π ⊗ π p ( G × G ) , ≤ p ≤ ∞ , associated to the tensor product representation π ⊗ π . The following result from [1, Theorem 3]shows that a kernel theorem is valid for general coorbit spaces. Theorem 3.46 (Coorbit Kernel Theorem) . Let π i : G i → U ( H i ) for i = 1 , be two integrablerepresentations. There is a bijective correspondence between bounded linear operators A : C o π ( G ) → C o π ∞ ( G ) and elements K ∈ C o ∞ ( G × G ) given by h Af, g i = h K, f ⊗ g i , (3.12) where f ∈ C o π ( G ) and g ∈ C o π ( G ) . Moreover, we have the norm-equivalence k A k Op ≍ k K k C o ∞ ( G × G ) . The reader is referred to [1, Section 5] for concrete applications of Theorem 3.46 regardingmappings between Besov spaces and modulation spaces. In light of Theorem 3.46, it makes senseto refer to K in (3.12) as the distributional kernel of the operator A : C o π ( G ) → C o π ∞ ( G ). Theauthors in [1] go on to use Theorem 3.46 to deduce properties of A based on knowledge of itsdistributional kernel K . In particular, they show the following elegant result in [1, Theorem 9]. Corollary 3.47.
In the notation of Theorem 3.46, the operator A defines a bounded linear operator A : C o π ∞ ( G ) → C o π ( G ) when its distributional kernel K satisfies K ∈ C o ( G × G ) . Now that all the main features of coorbit spaces have been discussed, we will briefly outline inSections 4.1 - 4.3 examples from different areas of modern analysis. The goal here is not to give acomprehensive exposition on each topic, nor to give a comprehensive account of all the applicationsof coorbit theory. We rather strive to convince the reader that coorbit theory is an active researchtopic that unifies seemingly different branches of modern analysis. We will in Sections 4.1 - 4.3provide references for further reading so that the reader can look more into the most eye-cachingexample themselves. Finally, in subsection 4.4 we give references to modern directions in coorbittheory, as well as suggestions for where the reader can learn more about coorbit theory.40 .1 Shearlet Spaces
For image analysis and image processing, the continuous wavelet transform given in (2.18) has beenextensively used. However, the continuous wavelet transform can fall short if one wishes to extractdirectional information. Several approaches have been developed to provide an alternative to thecontinuous wavelet transform, e.g. ridgelets and curvelets [6, 7, 68]. The most examined alternative,namely shearlets , does have a description that allows the theory of coorbit spaces to be applied.We refer the reader to [46, 55] for the origins of shearlets and to [54] for a general introduction toshearlets. In this section, we will describe the shearlet transform and the underlying shearlet groupin two dimensions following [15]. The extension to higher dimensions was given in [18].To begin describing the shearlet group we first need two matrices: For a ∈ R ∗ := R \ { } the parabolic scaling matrix A a is given by A a := a √ a ! , when a > , a −√− a ! , when a < . Hence A a for a > s ∈ R the shear matrix S s is given by S s := (cid:18) s (cid:19) . Using these matrices, we can define the shearlet group as follows.
Definition 4.1.
The (full) shearlet group S is defined to be R ∗ × R × R with the group operation( a, s, t ) · S ( a ′ , s ′ , t ′ ) := ( aa ′ , s + s ′ p | a | , t + S s A a t ′ ) . (4.1)It is straightforward to check that (4.1) is in fact a group operation with identity element(1 , , ∈ S , see [14, Lemma 2.1] for details. Notice that S has two connected components; theidentity component S + is called the connected shearlet group . The left Haar measure µ L on theshearlet group S is given by µ L ( a, s, t ) = da ds dt | a | , ( a, s, t ) ∈ S . Given an invertible matrix M ∈ GL (2 , R ) we can consider the generalized dilation operator D M acting on f ∈ L ( R ) by the formula D M f ( x ) := 1 p | det( M ) | f ( M − x ) , x ∈ R . (4.2)Notice that (4.2) is a two-dimensional generalization of the dilation operator given in (2.13). Definition 4.2.
The (continuous) shearlet representation π : S → U ( L ( R )) is given by π ( a, s, t ) f ( x ) := T t D S s A a f ( x ) = 1 p | det( S s A a ) | f (cid:0) A − a S − s ( x − t ) (cid:1) , where f ∈ L ( R ) and ( a, s, t ) ∈ S . 41ne can view the unitary representation π as a two-dimensional version of the continuous waveletrepresentation in (2.14). The representation π is irreducible since we are considering the full shearletgroup S instead of the connected group S + , see [15, Theorem 2.2]. Moreover, [15, Theorem 2.2] alsoshows that π is square integrable. More precisely, a function g ∈ L ( R ) is admissible if and only if Z R |F ( g )( ω , ω ) | ω dω dω = 1 . (4.3)We refer to the elements g ∈ L ( R ) satisfying (4.3) as (continuous) shearlets . Although it iscommon in the literature to denote the wavelet transform corresponding to the shearlet represen-tation by SH , we will stick with our predefined notation W for consistency. Hence for a shearlet g ∈ L ( R ) and f ∈ L ( R ) the orthogonality relation (2.10) shows that Z S |W g f ( a, s, t ) | µ L ( a, s, t ) = Z S |h f, π ( a, s, t ) g i| da ds dt | a | = k f k L ( R ) . Let us for simplicity consider the polynomial weights v α ( a, s, t ) := (1 + a + s ) α/ for ( a, s, t ) ∈ S and α ≥
0. The existence of a v α -integrable vector is guaranteed by [15, Theorem 4.2].Thus we obtain the space of v α -test vectors H v α and the v α -reservoir R /v α , see Section 3.5 fordetails. The following definition is inevitable. Definition 4.3.
The shearlet coorbit spaces C o p,v α ( S ) for 1 ≤ p ≤ ∞ are defined to be C o p,v α ( S ) := { f ∈ R /v α : W g f ∈ L pv α ( S ) } , where g is any v α -integrable vector.We invoke Theorem 3.34 to deduce that the shearlet coorbit spaces C o p,v α ( S ) constitute a well-behaved class of Banach spaces. In [15, Theorem 4.7] it is shown that the shearlet coorbit spacescontain many smooth functions of rapid decay. We refer the reader to [15, Section 4.2] for resultsregarding atomic decompositions and Banach frames for the shearlet coorbit spaces. We will now describe an application of coorbit spaces to the realm of classical complex analysis,namely the Bergman spaces. The connection with Bergman spaces was to our knowledge initiallypointed out in [28, Section 7.3]. To introduce this topic in a brief and succinct manner, we will givean outline of the definitions and results given in [57] and [31]. We encourage the reader to seek outthe more recent and technical paper [9] for interesting results in higher dimensions.Let us first recall the Bergman spaces in classical complex analysis. We denote the unit disk inthe complex plane by D and consider for α > − dA α ( z ) = α + 1 π (cid:0) − | z | (cid:1) α dz, z ∈ D . We let A pα := A pα ( D ) denote the (weighted) Bergman space consisting of analytic functions f : D → C such that Z D | f ( z ) | p dA α ( z ) < ∞ . p = 2 we have a natural Hilbert space structure on A α given by the inner product h f, g i α := Z D f ( z ) g ( z ) dA α ( z ) . It not difficult to verify that A α is a reproducing kernel Hilbert space with reproducing kernel K α ( z, w ) = 1(1 − zw ) α +2 , z, w ∈ D . We will now describe a group that acts unitarily on A α : For B := D × T we say that a functionon the form B a ( z ) := ǫ z − b − zb , z ∈ C , a := ( b, ǫ ) ∈ B , zb = 1 , is called a Blaschke function . The Blaschke functions allows us to define a group operation on B by the formula a ◦ a = a if and only if B a ◦ B a = B a . The locally compact group ( B , ◦ ) is aunimodular, non-commutative group known as the Blaschke group . Remark.
The terminology is motivated by the Blaschke products in complex analysis: A sequence( a n ) n ∈ N in D satisfies the Blaschke condition when ∞ X n =1 (1 − | a n | ) < ∞ . Given such a sequence, we define the
Blaschke product as the infinite product B ( z ) = ∞ Y n =1 B ( a n , z ) , B ( a, z ) = | a | a a − z − az , with the convention that B (0 , z ) = z . Then B is an analytic function in D vanishing precisely atthe points ( a n ) n ∈ N .Introduce the functions F a ( z ) := p ǫ (1 − | b | )1 − zb , a = ( b, ǫ ) ∈ B , z ∈ D . We obtain for each α ≥ U α : B → U ( A α ) given by U α ( a ) f ( z ) = [ F a − ( z )] α +2 f (cid:0) B − a ( z ) (cid:1) = [ F a − ( z )] α +2 f ( B a − ( z )) , f ∈ A α , a ∈ B , z ∈ D . The representation U α is square integrable and any g ∈ A α satisfying k g k α = π − √ α + 1 is admis-sible. For the wavelet transform W αg f ( a ) := h f, U α ( a ) g i α with f, g ∈ A α and g admissible we haveby Theorem 2.32 that W αg f = W αg f ∗ B W αg g. Moreover, we can by Corollary 2.34 reconstruct any f ∈ A α through the weak integral formula f ( z ) = Z B W αg f ( a )( U α ( a ) g )( z ) dµ L ( a )= 12 π Z π − π Z D W αg f ( b, e it )( U α ( b, e it ) g )( z )(1 − | b | ) db dt.
43 straightforward computation shows that for g ≡ ∈ A α we have kW αg g k L ( B ) = 2 /α . Hencefor α > U α is integrable. More generally, it is showed in[57, Theorem 3.2.2] that any non-zero analytic function g on the unit disk that can be written as g ( z ) = ∞ X j =0 λ j z − b j − zb j with | b j | ≤ j ≥ ∞ X j =0 | λ j | < ∞ , is an integrable vector for the representation U α for α >
0. For α > H α ⊂ A α and the reservoir space R α as usual, see Section 3.1 and 3.2 respectively fordetails. As such, we can define coorbit spaces associated to U α . Definition 4.4.
The
Blaschke coorbit spaces C o p,α ( B ) for 1 ≤ p ≤ ∞ and α > C o p,α ( B ) := { f ∈ R α : W αg f ∈ L p ( B ) } , where g is any integrable vector for U α .By the theory we have developed, we can automatically deduce all the consequences in Theorem3.34 for the Blaschke coorbit spaces C o p,α ( B ). For discretization results, the reader can first consult[57, Section 3.3] and proceed to [9, Theorem 3.14] where the classical atomic decomposition resultsfor Bergman spaces by Coifman and Rochberg are deduced through coorbit theory. It is clear from Example 2.3 that the modulation spaces are intrinsically linked with the Heisenberggroup. The Heisenberg group fits in with a large class of well-behaved locally compact groups knownas nilpotent Lie groups . We refer the reader to [33] for the definition of a nilpotent Lie group. Inview of this observation, it makes sense to try to define coorbit spaces analogous to the modulationspaces for other nilpotent groups. This is a recent idea that was first seriously considered in [32] andrecently expanded on in [44]. We will outline basic definitions and results in this direction following[44]. The interested reader should consult [32, 44] for more details and interesting examples.Let G be a simply connected nilpotent Lie group with center Z := Z ( G ) := { x ∈ G : xy = yx for all y ∈ G } . We will consider the quotient group G/ Z with its Haar measure µ G/ Z . An irreducible unitaryrepresentation π : G → U ( H π ) is said to be square integrable modulo the center if there exists anelement g ∈ H π such that Z G/ Z |W g g ( x ) | dµ G/ Z ( x ) < ∞ , (4.4)where as usual W g f ( x ) := h f, π ( x ) g i for f, g ∈ H π and x ∈ G . Since π | Z ( x ) = χ ( x ) · Id H π where χ is a character of the commutative group Z , it follows that the integrand in (4.4) is a well definedfunction on the quotient group G/ Z . We remind the reader that the reduction from G to the quotientgroup G/ Z is precisely what we did in Example 2.20 to make the Schr¨odinger representation squareintegrable. Hence we can say that the Schr¨odinger representation ρ : H n → U ( L ( R n )) given in(2.5) is square integrable modulo the center. 44o proceed, we first need a good choice for a well-behaved “window function” g ∈ H π . Since G is a Lie group it has a smooth structure and it makes sense to ask for a fixed g ∈ H π whether thefunction G ∋ x π ( x ) g ∈ H π (4.5)is a smooth map from G to H π . Details for this can be found in [33, Chapter 1.7]. We refer to theelements g ∈ H π such that (4.5) is a smooth map as the smooth vectors of the representation π anddenote them by H ∞ π . It is a general fact that H ∞ π is dense in H π , see [33, Proposition 1.7.7]. Definition 4.5.
Let G be a simply connected nilpotent Lie group with center Z . Assume wehave a square integrable representation modulo the center π : G → U ( H π ) and let H ∞ π denote thecorresponding smooth vectors. We define the coorbit space C o p ( G/ Z ) for 1 ≤ p < ∞ to be thecompletion of the subspace of elements f ∈ H ∞ π such that k f k C o p ( G/ Z ) = Z G/ Z |W g f ( x ) | p dµ G/ Z ( x ) ! p < ∞ , where g ∈ H ∞ π is a fixed non-zero smooth vector. Remark.
The case of C o ∞ ( G/ Z ) can be handled by considering weak closures, but we restrictourselves to 1 ≤ p < ∞ for simplicity. Moreover, we also refrain from considering weighted versionof C o p ( G/ Z ) so that we can focus on the essential features.Although the representation space H π has an abstract flavor in general, it can be shown thatfor nilpotent groups one can always realize H π as L ( R s ) in a natural way. We point out that theparameter s generally satisfies s < dim( G ). The identification of H π with L ( R s ) uses Kirillov’stheory of coadjoint orbits (not to be confused with coorbit theory). We refer the reader to thestandard reference [52, Chapter 3] for a more careful explanation of this phenomenon.One important problem for coorbit spaces on nilpotent groups is whether the new spaces areidentical to the classical modulation spaces. If this was the case, then coorbit spaces on nilpotentgroups would just be a more complicated view of the usual modulation spaces and offer little of value.The following example, taken from [44, Example 3.2], illustrates that this can actually happen. Example 4.6.
We consider the nilpotent group G with the concrete realization G ≃ ( R , · ) where x · y := ( x + y + x y + x y , x + y + x y , x + y , x + y , x + y , x + y ) . A square integrable representation modulo the center is π : G → U ( L ( R )) given by π ( x , . . . , x ) g ( s, t ) = e πi ( x − x s − x t ) g ( s − x , t − x ) = e πix M ( − x , − x ) T ( x ,x ) g ( s, t ) , where T and M are the translation operator and modulation operator given in (2.4). As our goalis to investigate the integrability of the corresponding wavelet transform, we henceforth drop thephase factor e πix as this will be insignificant. We identify G/ Z ≃ R and write x = (0 , , x , x , x , x ) ∈ G/ Z . The wavelet transform W g f for f ∈ L ( R ) and a non-zero g ∈ H ∞ π is given by W g f ( x ) = V g f (( x , x ) , ( − x , − x )) , where V g f is the STFT. From this it follows that for 1 ≤ p < ∞ we have C o p ( G/ Z ) = M p ( R ) since k f k C o p ( G/ Z ) ≃ k f k M p ( R ) .
45n light of the previous example, one might fear that coorbit spaces associated with nilpotentgroups never produce anything other than the classical modulation spaces. However, in [44] severalexamples are given of coorbit spaces on nilpotent groups that are not equal to any of the classicalmodulation spaces. The first example of this phenomenon was presented in [32, Theorem 7.6]. Thegroup in question was the
Dynin-Folland group , and the techniques used to prove distinctness camefrom the theory of decomposition spaces . Distinctness of a class of decomposition spaces on two-stepnilpotent groups was proved in [2, Theorem 5.6].
Phew! You’re still here? Good. Hopefully you have been convinced that coorbit theory is anexciting research topic. You now understand the main ideas of coorbit theory along with severalconcrete examples. If you are satisfied, then congratulations; you know the basics of coorbit theory.However, if you are interested in doing research in coorbit theory, then the journey has just started.A great way to get more familiar with technical aspects of coorbit theory is by reading the Ph.D.thesis of Felix Voigtlaender [63]. We also recommend seeking out the original papers on coorbittheory [28, 29, 30]. Reading these sources is will improve your fundamental knowledge of coorbittheory. A good idea is to find a problem in coorbit theory that you want to solve. This forces youto work through details that is tempting to skip when reading other peoples work. Below we havegiven some references for two directions that have received much attention in recent decades: • Consider two integrable representations π : G → U ( H ) and π : G → U ( H ) and twoparameters 1 ≤ p, q ≤ ∞ . A natural question to answer is whether there exists a continuousembedding φ : C o π p ( G ) → C o π q ( G )between different coorbit spaces corresponding to (possibly) different groups. This questionhas been considered in many concrete settings, see e.g. [23, 62] for the modulation spacesand Besov spaces, and [19] for embeddings between shearlet coorbit spaces. The embeddingquestion is often more easily tackled if the coorbit spaces in question can be given a decompo-sition space structure. Decomposition spaces originate from [27] and many general embeddingresults between decomposition spaces can be found in [65]. We refer the authors to [39] wherethe authors show that a large class of wavelet spaces can be given a decomposition spacestructure. In [64] several embedding results from decomposition spaces into Sobolev spacesand BV spaces are given. Specific embeddings between decomposition spaces with a geometricflavor have recently been investigated in [2, 4]. Finally, recent results regarding embeddingsof shearlet coorbit spaces into Sobolev spaces can be found in [37]. • There are plenty of directions where coorbit theory can be generalized: As previously men-tioned, one can instead of L p ( G ) for 1 ≤ p ≤ ∞ in the definition of C o p ( G ) consider C o ( Y ),where Y is a solid and translation invariant Banach space of functions on G , see [28, 29, 30].We refer the reader to [59, 63, 51] for results regarding coorbit spaces in the quasi-Banachsetting. The paper [12] considers coorbit spaces associated with representations that are notnecessarily integrable, while [38] considers certain representations that are not necessarilyirreducible. In [10] it is shown that atomic decompositions are valid even for projective rep-resentations. Coorbit theory for homogeneous spaces have been investigated, and we suggestto start with the papers [13, 15, 17]. We highly recommend the recent work [61] where theauthors derive discretization improvements and, in their own words, “bridge a gap betweenwhat is achievable with abstract and concrete methods”. Finally, we refer the reader to [35]46here a generalization of the coorbit space theory is used to derive atomic decompositionsand Banach frames for a wide range of Banach spaces.If you have found a typographical or mathematical error when reading this survey, it would bevery much appreciated if you would let me know. The same goes if some work on coorbit theoryyou believe deserves to be mentioned has been omitted. Department of Mathematical Sciences, Norwegian University of Science and Technology,7491 Trondheim, Norway.
E-mail addresses : [email protected] References [1] P. Balazs, K. Gr¨ochenig, and M. Speckbacher. Kernel theorems in coorbit theory.
Transactionsof the AMS, Series B , 6:346–364, 2019.[2] E. Berge. α -modulation spaces for step two stratified Lie groups. arXiv preprint: 1908.09567 ,2019.[3] E. Berge. Interpolation in wavelet spaces and the HRT-conjecture. To appear in: Journal ofPseudo-Differential Operators and Applications , 2020.[4] E. Berge and F. Luef. A large scale approach to decomposition spaces. arXiv preprint:1902.07797 , 2019.[5] F. Bruhat. Distributions sur un groupe localement compact et applications `a l’´etude desrepr´esentations des groupes p -adiques. Bulletin de la Soci´et´e Math´ematique de France , 89:43–75, 1961.[6] E. J. Cand`es and D. L. Donoho. Ridgelets: A key to higher-dimensional intermittency?
Philo-sophical Transactions of the Royal Society A , 357:2495–2509, 1999.[7] E. J. Cand`es and D. L. Donoho. Curvelets, multiresolution representation, and scaling laws.
Proceedings of SPIE - The International Society for Optical Engineering , 4119:1–12, 2000.[8] P. G. Casazza, D. Han, and D. Larson. Frames for Banach spaces.
Contemporary Mathematics ,247:149–182, 1999.[9] J. Christensen, K. Gr¨ochenig, and G. Olafsson. New atomic decompositons for Bergman spaceson the unit ball.
Indiana University Mathematics Journal , 66, 2015.[10] O. Christensen. Atomic decomposition via projective group representations.
Rocky MountainJournal of Mathematics , 26:1289–1312, 1996.[11] O. Christensen.
An Introduction to Frames and Riesz Bases, Second Edition . Birkh¨auser Basel,2016.[12] S. Dahlke, F. D. Mari, E. D. Vito, L. Sawatzki, G. Steidl, G. Teschke, and F. Voigtlaender.On the atomic decomposition of coorbit spaces with non-integrable kernel. In
Landscapes ofTime-Frequency Analysis , pages 75–144. Birkh¨auser, 2019.4713] S. Dahlke, M. Fornasier, H. Rauhut, G. Steidl, and G. Teschke. Generalized coorbit theory, Ba-nach frames, and the relation to α -modulation spaces. Proceedings of the London MathematicalSociety , 96:464–506, 2008.[14] S. Dahlke, G. Kutyniok, P. Maass, C. Sagiv, H. Stark, and G. Teschke. The uncertaintyprinciple associated with the continuous shearlet transform.
International Journal of Wavelets,Multiresolution and Information Processing , 6:157–181, 2008.[15] S. Dahlke, G. Kutyniok, G. Steidl, and G. Teschke. Shearlet coorbit spaces and associatedBanach frames.
Applied and Computational Harmonic Analysis , 27:195–214, 2009.[16] S. Dahlke, F. D. Mari, P. Grohs, and D. Labate.
Harmonic and Applied Analysis: From Groupsto Signals . Birkh¨auser, 2015.[17] S. Dahlke, G. Steidl, and G. Teschke. Frames and coorbit theory on homogeneous spaces witha special guidance on the sphere.
Journal of Fourier Analysis and Applications , 13:387–403,2007.[18] S. Dahlke, G. Steidl, and G. Teschke. The continuous shearlet transform in arbitrary spacedimensions.
Journal of Fourier Analysis and Applications , 16:340–364, 2010.[19] S. Dahlke, G. Steidl, and G. Teschke. Shearlet coorbit spaces: compactly supported analyzingshearlets, traces and embeddings.
Journal of Fourier Analysis and Applications , 17:1232–1255,2011.[20] I. Daubechies.
Ten Lectures on Wavelets . SIAM, 1992.[21] A. Deitmar and S. Echterhoff.
Principles of Harmonic Analysis, Second Edition . Springer,2014.[22] M. Duflo and C. C. Moore. On the regular representation of a nonunimodular locally compactgroup.
Journal of Functional Analysis , 21:209–243, 1976.[23] D. Fan, W. Guo, and G. Zhao. Full characterization of the embedding relations between α -modulation spaces. Science China Mathematics , 61:1243–1272, 2018.[24] H. G. Feichtinger. Banach convolution of Wiener type. In
Functions, Series, Operators, Proc.Conf. Budapest , volume 38, pages 509–524, 1980.[25] H. G. Feichtinger. On a new Segal algebra.
Monatshefte f¨ur Mathematik , 92:269–289, 1981.[26] H. G. Feichtinger. Banach spaces of distributions defined by decomposition methods. II.
Math.Nachr , 132:207–237, 1987.[27] H. G. Feichtinger and P. W. Gr¨obner. Banach spaces of distributions defined by decompositionmethods. I.
Mathematische Nachrichten , 123:97–120, 1985.[28] H. G. Feichtinger and K. Gr¨ochenig. A unified approach to atomic decompositions via integrablegroup representations. In
Function Spaces and Applications , pages 52–73. Springer, 1988.[29] H. G. Feichtinger and K. Gr¨ochenig. Banach spaces related to integrable group representationsand their atomic decompositions, part I.
Journal of Functional analysis , 86:307–340, 1989.4830] H. G. Feichtinger and K. Gr¨ochenig. Banach spaces related to integrable group representationsand their atomic decompositions, part II.
Monatshefte f¨ur Mathematik , 108:129–148, 1989.[31] H. G. Feichtinger and M. Pap. Coorbit theory and Bergman spaces. In
Harmonic and ComplexAnalysis and its Applications , pages 231–259. Springer, 2014.[32] V. Fischer, D. Rottensteiner, and M. Ruzhansky. Heisenberg-modulation spaces at the cross-roads of coorbit theory and decomposition space theory. arXiv preprint: 1812.07876 , 2018.[33] V. Fischer and M. Ruzhansky.
Quantization on Nilpotent Lie Groups . Springer Nature, 2016.[34] G. B. Folland.
A Course in Abstract Harmonic Analysis, Second Edition . Chapman andHall/CRC, 2016.[35] M. Fornasier and H. Rauhut. Continuous frames, function spaces, and the discretization prob-lem.
Journal of Fourier Analysis and Applications , 11:245–287, 2005.[36] H. F¨uhr.
Abstract Harmonic Analysis of Continuous Wavelet Transforms . Springer, 2005.[37] H. F¨uhr and R. Koch. Embeddings of shearlet coorbit spaces into Sobolev spaces.
InternationalJournal of Wavelets, Multiresolution and Information Processing , 2020.[38] H. F¨uhr and J. T. Van Velthoven. Coorbit spaces associated to integrably admissible dilationgroups.
To appear in: Journal d’Analyse Math´ematique , 2020.[39] H. F¨uhr and F. Voigtlaender. Wavelet coorbit spaces viewed as decomposition spaces.
Journalof Functional Analysis , 269:80–154, 2015.[40] M. Ghandehari and K. F. Taylor. Images of the continuous wavelet transform. In
OperatorMethods in Wavelets, Tilings, and Frames , pages 55–65. American Mathematical Society, 2014.[41] K. Gr¨ochenig. Describing functions: Atomic decompositions versus frames.
Monatshefte f¨urMathematik , 112:1–42, 1991.[42] K. Gr¨ochenig.
Foundations of Time-Frequency Analysis . Springer Science & Business Media,2001.[43] K. Gr¨ochenig. Time-frequency analysis of Sj¨ostrand’s class.
Revista Matem´atica Iberoameri-cana , 22:703–724, 2006.[44] K. Gr¨ochenig. New function spaces associated to representations of nilpotent Lie groups andgeneralized time-frequency analysis. arXiv preprint: 2007.04615 , 2020.[45] A. Grossmann, J. Morlet, and T. Paul. Transforms associated to square integrable grouprepresentations. I. General results.
Journal of Mathematical Physics , 26:2473–2479, 1985.[46] K. Guo, G. Kutyniok, and D. Labate. Sparse multidimensional representations usinganisotropic dilation and shear operators.
In Proceedings of the International Conference onthe Interactions between Wavelets and Splines, Athens , 2005.[47] C. Heil. An introduction to weighted Wiener amalgams. In
Allied Publishers, New Delhi. ,pages 183–216. Citeseer, 2003.[48] E. Hewitt and K. A. Ross.
Abstract Harmonic Analysis: Volume I Structure of TopologicalGroups Integration Theory Group Representations . Springer Science & Business Media, 2012.4949] R. Howe. On the role of the Heisenberg group in harmonic analysis.
Bulletin of the AmericanMathematical Society , 3:821–843, 1980.[50] M. S. Jakobsen. On a (no longer) new Segal algebra: a review of the Feichtinger algebra.
Journal of Fourier Analysis and Applications , 24:1579–1660, 2018.[51] H. Kempka, M. Sch¨afer, and T. Ullrich. General coorbit space theory for quasi-Banach spacesand inhomogeneous function spaces with variable smoothness and integrability.
Journal ofFourier Analysis and Applications , 23:1348–1407, 2017.[52] A. A. Kirillov.
Lectures on the Orbit Method . American Mathematical Soc., 2004.[53] R. Koch.
Analysis of shearlet coorbit spaces . PhD thesis, RWTH Aachen, 2018.[54] G. Kutyniok and D. Labate. Introduction to shearlets. In
Shearlets , pages 1–38. Springer,2012.[55] G. Kutyniok, D. Labate, W. Lim, and G. Weiss. Sparse multidimensional representation usingshearlets. In
Wavelets XI , volume 5914. Proceedings of SPIE - The International Society forOptical Engineering, 2005.[56] F. Luef and E. Skrettingland. A Wiener Tauberian theorem for operators and functions. arXivpreprint: 2005.04160 , 2020.[57] M. Pap. Properties of the voice transform of the Blaschke group and connections with atomicdecomposition results in the weighted Bergman spaces.
Journal of Mathematical Analysis andApplications , 389:340–350, 2012.[58] V. I. Paulsen and M. Raghupathi.
An Introduction to the Theory of Reproducing Kernel HilbertSpaces . Cambridge University Press, 2016.[59] H. Rauhut. Coorbit space theory for quasi-Banach spaces.
Studia Mathematica , 180:237–253,2005.[60] A. Robert.
Introduction to the Representation Theory of Compact and Locally Compact Groups .Cambridge University Press, 1983.[61] J. L. Romero, J. T. Van Velthoven, and F. Voigtlaender. On dual molecules and convolution-dominated operators. arXiv preprint: 2001.09609 , 2020.[62] J. Toft and P. Wahlberg. Embeddings of α -modulation spaces. Pliska Studia MathematicaBulgarica , 21:25–56, 2012.[63] F. Voigtlaender.
Embedding theorems for decomposition spaces with applications to waveletcoorbit spaces . PhD thesis, RWTH Aachen University, 2016.[64] F. Voigtlaender. Embeddings of decomposition spaces into Sobolev and BV spaces. arXivpreprint: 1601.02201 , 2016.[65] F. Voigtlaender. Embeddings of decomposition spaces. arXiv preprint: 1605.09705 , 2019.[66] E. Wilczok. New uncertainty principles for the continuous Gabor transform and the continuouswavelet transform.
Documenta Mathematica , 5:201–226, 2000.5067] M. W. Wong.
Wavelet Transforms and Localization Operators . Springer Science & BusinessMedia, 2002.[68] B. Zhang, J. M. Fadili, and J. Starck. Wavelets, ridgelets, and curvelets for Poisson noiseremoval.