Co-compact Gabor systems on locally compact abelian groups
aa r X i v : . [ m a t h . F A ] A p r Co-compact Gabor systems on locally compact abelian groups
Mads Sielemann Jakobsen ∗ , Jakob Lemvig † April 15, 2015
Abstract:
In this work we extend classical structure and duality results in Gaboranalysis on the euclidean space to the setting of second countable locally compactabelian (LCA) groups. We formulate the concept of rationally oversampling ofGabor systems in an LCA group and prove corresponding characterization resultsvia the Zak transform. From these results we derive non-existence results for crit-ically sampled continuous Gabor frames. We obtain general characterizations intime and in frequency domain of when two Gabor generators yield dual frames.Moreover, we prove the Walnut and Janssen representation of the Gabor frameoperator and consider the Wexler-Raz biorthogonality relations for dual genera-tors. Finally, we prove the duality principle for Gabor frames. Unlike most dualityresults on Gabor systems, we do not rely on the fact that the translation and mod-ulation groups are discrete and co-compact subgroups. Our results only rely onthe assumption that either one of the translation and modulation group (in somecases both) are co-compact subgroups of the time and frequency domain. Thispresentation offers a unified approach to the study of continuous and the discreteGabor frames.
In Gabor analysis structure and duality results, such as the Zibulski-Zeevi, the Walnut and theJanssen representation of the frame operator, the Wexler-Raz biorthogonal relations, and theduality principle, play an important role. These results go back to a series of papers in the1990s [14, 30–32, 39–43, 45–47] on (discrete) regular Gabor systems in L ( R ) and L ( R n ) withmodulations and translations along full-rank lattices. The results now constitute a fundamentalpart of the theory. In L ( R n ) , a regular Gabor system is a discrete family of functions of theform { E γ T λ g } λ ∈ A Z n ,γ ∈ B Z n , where g ∈ L ( R n ) , E γ T λ g ( x ) = e πiγ · x g ( x − λ ) , and A, B ∈ GL n ( R ) .For Gabor systems on locally compact abelian (LCA) groups, the picture is a lot less complete.Rieffel [38] proved in 1988 a weak form of the Janssen representation called the fundamentalidentity in Gabor analysis (FIGA) for Gabor systems in L ( G ) with modulations and translationsalong a closed subgroup in G × b G , where G is a second countable LCA group and b G its dual group.Most other structure and duality results assume Gabor systems in L ( G ) with modulations andtranslations along discrete and co-compact subgroups (also called uniform lattices), e.g., theWexler-Raz biorthogonal relations for such uniform lattice Gabor systems appear implicitly in Mathematics Subject Classification.
Primary 42C15, Secondary: 43A32, 43A70.
Key words and phrases.
Dual frames, duality principle, fiberization, frame, Gabor system, Janssen represen-tation, LCA group, Walnut representation, Wexler-Raz biorthogonality relations, Zak transform ∗ Technical University of Denmark, Department of Applied Mathematics and Computer Science, Matematik-torvet 303B, 2800 Kgs. Lyngby, Denmark, E-mail: [email protected] † Technical University of Denmark, Department of Applied Mathematics and Computer Science, Matematik-torvet 303B, 2800 Kgs. Lyngby, Denmark, E-mail: [email protected] akobsen, Lemvig Co-compact Gabor systems on LCA groups the work of Gröchenig [23]. Uniform lattices are discrete subgroups whose quotient group iscompact, and thus, they are natural generalizations of the concept of full-rank lattices in R n .However, not all LCA groups possess uniform lattices. This naturally leads to the questionto what extent the classical results on Gabor theory mentioned above can be formulated fornon-lattice Gabor systems. The current paper gives an answer to this question.Thus, in this work we set out to extend the theory of structure and duality results to alarge class of Gabor systems in L ( G ) , where G is a second countable LCA group. We willfocus on so-called co-compact Gabor systems { E γ T λ g } λ ∈ Λ ,γ ∈ Γ , where translation and modu-lation of g ∈ L ( G ) are along closed, co-compact (i.e., the quotient group is compact) sub-groups Λ ⊂ G and Γ ⊂ b G , respectively. In L ( R n ) co-compact Gabor systems are of the form (cid:8) e πiγ · x g ( x − λ ) (cid:9) λ ∈ A ( R s × Z d − s ) ,γ ∈ B ( R r × Z d − r ) for some choice of ≤ r, s ≤ d . Depending on theparameters r and s , these Gabor systems range from discrete over semi-continuous to continuousfamilies. If only one of the subsets Λ and Γ is a closed, co-compact subgroup, we will use theterminology semi co-compact Gabor system. Clearly, co-compact and semi co-compact Gaborsystems need not be discrete. More importantly, such systems exist for all LCA groups, and thissetup unifies discrete and continuous Gabor theory.For co-compact Gabor systems we prove Walnut’s representation (Theorem 5.5) and Janssen’srepresentation (Theorem 5.7) of the Gabor frame operator, the Wexler-Raz biorthogonal relations(Theorem 6.5), and the duality principle (Theorem 6.7). As an example, we mention that thisgeneralized duality principle for L ( R n ) says that the co-compact Gabor system n e πiγ · x g ( x − λ ) : λ ∈ A ( R s × Z d − s ) , γ ∈ B ( R r × Z d − r ) o is a continuous frame if, and only if, the adjoint system n e πiγ · x g ( x − λ ) : λ ∈ ( B T ) − ( { } r × Z d − r ) , γ ∈ ( A T ) − ( { } s × Z d − s ) o is a Riesz sequence. We recall that a family of vectors { f k } k ∈ M in a Hilbert space H is acontinuous frame with respect to a measure µ on the index set M if k f k ≍ R M |h f, f k i| dµ forall f ∈ H and that { g k } k ∈ N is a Riesz sequence if k c k ℓ ≍ k P k c k g k k for all finite sequences c = { c k } k ∈ N . Our proof of the duality principle relies on a simple characterization of Rieszsequences in Hilbert spaces (Theorem 6.6).As we will see, the setting of co-compact Gabor systems is indeed a natural framework forstructure and duality results. Closedness of the modulation and translation subgroups is a stan-dard assumption, and one cannot get very far without it, e.g., closedness allows for applicationsof key identifications between subgroups and their annihilators as well as applications of the Weiland the Poisson formulas. Co-compactness is, on the other hand, non-standard, and to the bestof our knowledge this work is the first systematic study of co-compact Gabor systems. Underthe second countability assumption on G , co-compactness is the weakest assumption that yieldsan adjoint Gabor system with modulations and translations along discrete and countable sub-groups. In this way, co-compactness of the respective subgroups is the most general setting forwhich the Wexler-Raz biorthogonal relations, the duality conditions for dual generators and theduality principle can be phrased in a way that resembles the classical statements in L ( R n ) . Asan example we mention that in the Wexler-Raz biorthogonal relations, one characterize dualityof two Gabor frames by a biorthogonality condition of the corresponding adjoint Gabor systems.Since L ( G ) is separable, such a biorthogonality condition is only possible if the adjoint systemsare countable sequences (which co-compactness exactly guarantees). Furthermore, co-compactGabor systems are precisely the setting, where the Walnut and Janssen representation of thecontinuous frame operator are a discrete representation. akobsen, Lemvig Co-compact Gabor systems on LCA groups However, we begin our work on Gabor systems with a study of semi co-compact Gaborsystems as special cases of co-compact translation invariant systems, recently introduced in[7, 29]. For translation invariant systems we consider fiberization characterization of frames fortranslation invariant subspaces (Theorem 3.1), generalizing results from [5, 7, 8, 39]. Using thesefiberization techniques we will develop Zak transform methods for Gabor analysis in L ( G ) . Thisleads among other things to a concept of rational oversampling in LCA groups (Theorem 4.3)and a Zibulski-Zeevi representation (Corollary 4.4). Furthermore, we will prove the non-existenceof continuous, semi co-compact Gabor frames at “critical density” (Theorem 4.2). We also givecharacterizations of generators of dual semi co-compact Gabor frames (Theorems 4.7 and 4.9).There are several advantages of the LCA group approach, one being that the essential ingre-dient in our arguments often becomes more transparent than in the special cases. The abstractapproach also allows us to unify results from the standard settings where G is usually R n , Z n ,or Z n . This is not only useful for the sake of generalizations, but, in some instances, it can alsosimplify the proofs in the special cases. As an example we mention that our proof of the Zaktransform characterization of Gabor frames is based on two applications of the same result onfiberizations of L ( G ) , but for two different LCA groups G . In the Euclidean setting this wouldrequire two different fiberization results, one for G = R n and one for G = A Z n for A ∈ GL n ( R ) .In the setting of LCA groups we can unify such results into one general result. On the otherhand, even for G = R n most of our results are new.For related work on locally compact (abelian) groups we refer to the recent papers [2, 3, 7, 8,13, 18, 29, 34] as well as the book [20] and the references therein.The paper is organized as follows. In Section 2 we give a brief introduction to harmonicanalysis on LCA groups and frame theory. In Section 3 we study co-compact translation invariantsystems, and specialize to semi co-compact Gabor systems in Section 4. In Section 5 we studythe frame operator of Gabor systems, and in Section 6 we present duality results on co-compactGabor frames. In the following sections we set up notation and recall some useful results from Fourier analysison locally compact abelian groups and continuous frame theory.
In this paper G will denote a second countable locally compact abelian group. To G we associateits dual group b G which consists of all characters, i.e., all continuous homomorphisms from G into the torus T ∼ = { z ∈ C : | z | = 1 } . Under pointwise multiplication b G is also a locally compactabelian group. Throughout the paper we use addition and multiplication as group operation in G and b G , respectively. By the Pontryagin duality theorem, the dual group of b G is isomorphic to G as a topological group, i.e., bb G ∼ = G . Moreover, if G is discrete, then b G is compact, and if G iscompact, then b G is discrete.We denote the Haar measure on G by µ G . The (left) Haar measure on any locally compactgroup is unique up to a positive constant. From µ G we define L ( G ) and the Hilbert space L ( G ) over the complex field in the usual way. L ( G ) is separable, because G is assumed to be secondcountable. For functions f ∈ L ( G ) we define the Fourier transform F f ( ω ) = ˆ f ( ω ) = Z G f ( x ) ω ( x ) dµ G ( x ) , ω ∈ b G. If f ∈ L ( G ) , ˆ f ∈ L ( b G ) , and the measure on G and b G are normalized so that the Planchereltheorem holds (see [27, (31.1)]), the function f can be recovered from ˆ f by the inverse Fourier akobsen, Lemvig Co-compact Gabor systems on LCA groups transform f ( x ) = F − ˆ f ( x ) = Z b G ˆ f ( ω ) ω ( x ) dµ b G ( ω ) , a.e. x ∈ G. We assume that the measure on a group µ G and its dual group µ b G are normalized this way, andwe refer to them as dual measures . We will consider F as an isometric isomorphism between L ( G ) and L ( b G ) .On any locally compact abelian group G , we define the following three operators. For a ∈ G ,the operator T a , called translation by a , is defined by T a : L ( G ) → L ( G ) , ( T a f )( x ) = f ( x − a ) , x ∈ G. For χ ∈ b G , the operator E χ , called modulation by χ , is defined by E χ : L ( G ) → L ( G ) , ( E χ f )( x ) = χ ( x ) f ( x ) , x ∈ G. For t ∈ L ∞ ( G ) the operator M t , called multiplication by t , is defined by M t : L ( G ) → L ( G ) , ( M t f )( x ) = t ( x ) f ( x ) , x ∈ G. The following commutator relations will be used repeatedly: T a E χ = χ ( a ) E χ T a , F T a = E a − F ,and F E χ = T χ F . For a subset H of an LCA group G , we define its annihilator as A ( b G, H ) = { ω ∈ b G | ω ( x ) = 1 for all x ∈ H } . When the group b G is understood from the context, we will simply denote the annihilator A ( b G, H ) = H ⊥ . The annihilator is a closed subgroup in b G , and if H is a closed subgroupitself, then b H ∼ = b G/H ⊥ and [ G/H ∼ = H ⊥ . These relations show that for a closed subgroup H thequotient G/H is compact if and only if H ⊥ is discrete. Lemma 2.1.
Let H be a closed subgroup of G . If G/H is finite, then H ⊥ ∼ = G/H .Proof.
Note that any finite group G is self-dual, that is, b G ∼ = G . And so, by application of theisomorphism H ⊥ ∼ = [ G/H we find that H ⊥ ∼ = [ G/H ∼ = G/H .We also remind the reader of Weil’s formula; it relates integrable functions over G withintegrable functions on the quotient space G/H when H is a closed normal subgroup of G . Fora closed subgroup H of G we Let π H : G → G/H, π H ( x ) = x + H be the canonical map from G onto G/H . If f ∈ L ( G ) , then the function ˙ x R H f ( x + h ) dµ H ( h ) , ˙ x = π H ( x ) defined almosteverywhere on G/H , is integrable. Furthermore, when two of the Haar measures on
G, H and
G/H are given, then the third can be normalized such that Z G f ( x ) dx = Z G/H Z H f ( x + h ) dµ H ( h ) dµ G/H ( ˙ x ) . (2.1)Hence, if two of the measures on G, H, G/H, b G, H ⊥ and b G/H ⊥ are given, and these two are notdual measures, then by requiring dual measures and Weil’s formula (2.1), all other measures areuniquely determined. To ease notation, we will often write dh in place of dµ H ( h ) and likewisefor other measures.A Borel section or a fundamental domain of a closed subgroup H in G is a Borel measurablesubset X of G which meets each coset G/H once. Any closed subgroup H in G has a Borel akobsen, Lemvig Co-compact Gabor systems on LCA groups section [35, Lemma 1.1]; however, we shall in the following usually only consider Borel sectionsof discrete subgroups H . We always equip Borel sections of G with the Haar measure µ G | X .Assume that H is a discrete subgroup. It follows that µ G ( X ) is finite if, and only if, H is co-compact, i.e., H is a uniform lattice [7]. From [7], we also have that the mapping x x + H from ( X, µ G ) to ( G/H, µ
G/H ) is measure-preserving, and the mapping Q ( f ) = f ′ defined by f ′ ( x + H ) = f ( x ) , x + H ∈ G/H, x ∈ X, (2.2)is an isometry from L ( X, µ G ) onto L ( G/H, µ
G/H ) .For more information on harmonic analysis on locally compact abelian groups, we refer thereader to the classical books [22, 26, 27, 37]. One of the central concept of this paper is that of a frame. The definition is as follows.
Definition 2.2.
Let H be a complex Hilbert space, and let ( M, Σ M , µ M ) be a measure space,where Σ M denotes the σ -algebra and µ M the non-negative measure. A family of vectors { f k } k ∈ M is called a frame for H with respect to ( M, Σ M , µ M ) if(a) the mapping M → C , k
7→ h f, f k i is measurable for all f ∈ H , and(b) there exists constants A, B > such that A k f k ≤ Z M |h f, f k i| dµ M ( k ) ≤ B k f k for all f ∈ H . (2.3)The constants A and B are called frame bounds .If { f k } k ∈ M is measurable and the upper bound in the above inequality (2.3) holds, then { f k } k ∈ M is said to be a Bessel system or family with constant B . A frame { f k } k ∈ M is said to be tight if we can choose A = B ; if, furthermore, A = B = 1 , then { f k } k ∈ M is said to be a Parsevalframe .If µ M is the counting measure and Σ M = 2 M the discrete σ -algebra, we say that { f k } k ∈ M is a discrete frame whenever (2.3) is satisfied; for this measure space, any family of vectorsis obviously measurable. Because the results of the present paper can be formulated for thediscrete and continuous setting, we shall refer to either cases as frames and be more specificwhen necessary . We mention that in the literature frames and discrete frames are usually calledcontinuous frames and frames, respectively. The concept of continuous frames was introducedby Kaiser [33] and Ali, Antoine, and Gazeau [1]. For an introduction to frame theory, we referthe reader to [11].To a Bessel family { f k } k ∈ M for H , we associate the the synthesis operator T : L ( M, µ M ) →H defined weakly by T { c k } k ∈ M = Z M c k f k µ M ( k ) . (2.4)This is a bounded linear operator. Its adjoint operator T ∗ : H → L ( M, µ M ) is called the analysis operator , and it is given by T ∗ f = {h f, f k i} k ∈ M . (2.5)The frame operator S : H → H is then defined as S = T T ∗ . We remark that the frame operatoris the unique operator satisfying h Sf, g i = Z M h f, f k ih f k , g i dµ M ( k ) for all f, g ∈ H , (2.6) akobsen, Lemvig Co-compact Gabor systems on LCA groups and that it is well-defined, bounded and self-adjoint for any Bessel system { f k } k ∈ M ; it is invertibleif { f k } k ∈ M is a frame.In case the frame inequalities (2.3) only hold for f ∈ K := span { f k } k ∈ M ⊂ H , we say that { f k } k ∈ M is a basic frame or a frame for its closed linear span. For discrete frames such framesare usually called frame sequences; we will not adopt this terminology as basic frames need notbe sequences. A frame for H is clearly a basic frame with K = H . If we need to stress that abasic frame spans all of H , we use the terminology total frame . Now, let us briefly comment onthe definition of the subspace K .From the Bessel property of a (basic) frame { f k } , we see that: im T = (ker T ∗ ) ⊥ = { f ∈ H : h f, f k i = 0 ∀ k ∈ M } ⊥ = span { f k } k ∈ M . The lower frame bound for f ∈ K implies that the operator T ∗ | K is bounded from below, i.e., k T ∗ | K f k ≥ √ A k f k , which is equivalent to T ∗ | K being injective with closed range which, in turn,implies that T has closed range. Since T ∗ | K is injective, the range of T is dense in K . It followsthat im T = K .We will only consider measures µ M that are σ -finite. Assume that { f k } is measurable. It isknown that T as in (2.4) defines a bounded linear operator if, and only if, { f k } k ∈ M is a Besselfamily [36]. Hence, the argument in the preceding paragraph shows that { f k } k ∈ M is a basicframe if, and only if, T as in (2.4) defines a bounded linear operator with im T = K .Two Bessel systems { f k } k ∈ M and { g k } k ∈ M are said to be dual frames for H if h f, g i = Z M h f, g k ih f k , g i dµ M ( k ) for all f, g ∈ H . (2.7)In this case f = Z M h f, g k i f k dµ M ( k ) for f ∈ H , (2.8)holds in the weak sense. For discrete frames, equation (2.8) holds in the usual strong sense, i.e.,with (unconditional) convergence in the H norm. Two dual frames are indeed frames. We alsomention that to a given frame for H one can always find at least one dual frame, the so-calledcanonical dual frame { S − f k } k ∈ M .Let us end this section with the definition of a Riesz sequence. Definition 2.3.
Let { f k } ∞ k =1 be a sequence in a Hilbert space H . If there exists constants A, B > such that A X k | c k | ≤ (cid:13)(cid:13)(cid:13) X k c k f k (cid:13)(cid:13)(cid:13) H ≤ B X k | c k | for all finite sequence { c k } ∞ k =1 , then we call { f k } ∞ k =1 a Riesz sequence . If furthermore span { f k } ∞ k =1 = H , then { f k } ∞ k =1 is a Riesz basis . Before we focus on Gabor systems, let us first show some results concerning the class of translationinvariant systems, recently introduced in [7, 29], which contains the class of (semi) co-compactGabor systems.We define translation invariant systems as follows. Let P be a countable or an uncountableindex set, let g p ∈ L ( G ) for p ∈ P , and let H be a closed, co-compact subgroup in G . For acompact abelian group, the group is metrizable if, and only if, the character group is countable akobsen, Lemvig Co-compact Gabor systems on LCA groups [26, (24.15)]. Hence, since G/H is compact and metrizable, the group [ G/H ∼ = H ⊥ is discreteand countable. Unless stated otherwise we equip H ⊥ with the counting measure and assume afixed Haar measure µ G on G . The (co-compact) translation invariant (TI) system generated by { g p } p ∈ P with translationalong the closed, co-compact subgroup H is the family of functions { T h g p } h ∈ H,p ∈ P . We will usethe following standing assumptions on the index set P :(I) ( P, Σ P , µ P ) is a σ -finite measure space,(II) p g p , ( P, Σ P ) → ( L ( G ) , B L ( G ) ) is measurable,(III) ( p, x ) g p ( x ) , ( P × G, Σ P ⊗ B G ) → ( C , B C ) is measurable.We say that { g p } p ∈ P is admissible or, when g p is clear from the context, simply that the measurespace P is admissible. The nature of these assumptions are discussed in [29]. Observe that anyclosed subgroup P of G (or b G ) with the Haar measure is admissible if p → g p is continuous, e.g.,if g p = T p g for some function g ∈ L ( G ) .If P is countable, we equip it with a weighted counting measure. If the subgroup H is alsodiscrete, hence a uniform lattice, the system { T h g p } h ∈ H,p ∈ P is a shift invariant (SI) system. TI systems are of interest to us since the Gabor systems we shall study are special instances ofthese. As the work of Ron and Shen [39] and Bownik [5] show, certain Gramian and so-calleddual Gramian matrices as well as a fiberization technique play an important role in the studyof TI systems. The fiberization technique is closely related to Zak transform methods in Gaboranalysis, as we will see in Section 4.1.Let Ω ⊂ b G be a Borel section of H ⊥ in b G as defined in Section 2.1. Following [7] we definethe fiberization mapping T : L ( G ) → L (Ω , ℓ ( H ⊥ )) by T f ( ω ) = { ˆ f ( ωα ) } α ∈ H ⊥ , ω ∈ Ω; (3.1)the inner product in L (Ω , ℓ ( H ⊥ )) is defined in the obvious manner. Fiberization is an isometric,isomorphic operation as shown in [7, 8].Our first result characterizes the frame/Bessel property of TI systems in terms of fibers. Itextends results from [5, 7, 8] to the case of uncountable many generators { g p } p ∈ P . Theorem 3.1.
Let < A ≤ B < ∞ , let H ⊂ G be a closed, co-compact subgroup, and let { g p } p ∈ P ⊂ L ( G ) , where ( P, µ P ) is an admissible measure space. The following assertions areequivalent:(i) The family { T h g p } h ∈ H,p ∈ P is a frame for L ( G ) with bounds A and B (or a Bessel systemwith bound B ),(ii) For almost every ω ∈ Ω , the family {T g p ( ω ) } p ∈ P is a frame for ℓ ( H ⊥ ) with bounds A and B (or a Bessel system with bound B ).Proof. The proof follows from the proofs in [5, 7, 8]. Indeed, the key computation in [7] showsthat Z P Z H (cid:12)(cid:12) h f, T h g p i L (cid:12)(cid:12) dµ H ( h ) dµ P ( p ) = Z P Z Ω (cid:12)(cid:12) hT f ( ω ) , T g p ( ω ) i ℓ (cid:12)(cid:12) dµ b G ( ω ) dµ P ( p ) akobsen, Lemvig Co-compact Gabor systems on LCA groups for all f ∈ L ( G ) . Let us outline the argument for the frame case; the Bessel case is similar.Assume that (ii) holds. Then for a.e. ω ∈ Ω we have A k a k ℓ ≤ Z P (cid:12)(cid:12) h a, T g p ( ω ) i ℓ (cid:12)(cid:12) dµ P ( p ) ≤ B k a k ℓ for all a ∈ ℓ ( H ⊥ ) . If we integrate these inequalities over Ω and use that T is an isometric isomorphism, we arriveat (i) using the key computation above. The other implication follows as in [5]. Remark . Theorem 3.1 can also be formulated for basic frames using the notion of range func-tions. A very general version of this result was obtained independently and concurrently in [28].Theorem 3.1 is closely related to the theory of translation invariant subspaces which very recentlyhas been studied in [4, 28] using Zak transform methods (cf. Section 4.1).Theorem 3.1 shows that the task of verifying that a given TI system { T h g p } h ∈ H,p ∈ P is aframe for L ( G ) can be replaced by the simpler task of proving that the fibers {T g p ( ω ) } p ∈ P area frame for the discrete space ℓ ( H ⊥ ) , however, this needs to be done for every ω ∈ Ω . For auniform lattice H , the Borel section Ω of H ⊥ is compact, but for non-discrete, co-compact closedsubgroups H , this is not the case, in fact, m b G (Ω) = ∞ .Let ω ∈ Ω be given. The analysis operator L ω : ℓ ( H ⊥ ) → L ( P ) for the family of fibers {T g p ( ω ) } p ∈ P in ℓ ( H ⊥ ) is given by: L ω c = p
7→ h c, T g p ( ω ) i ℓ ( H ⊥ ) , D ( L ω ) = c ( H ⊥ ) . (3.2)Note that we have only defined the analysis operator L ω for finite sequences since we do not, apriori, assume that the family of fibers is a Bessel system, cf. (2.5). If L ω is bounded, it extendsto a bounded, linear operator on all of ℓ ( H ⊥ ) ; clearly, L ω is bounded with bound k L ω k ≤ √ B if, and only if, {T g p ( ω ) } p ∈ P is a Bessel system with bound B . In this case the adjoint is thesynthesis operator L ∗ ω : L ( P ) → ℓ ( H ⊥ ) given by: L ∗ ω f = (cid:26)Z P f ( p ) ˆ g p ( ωα ) dµ P ( p ) (cid:27) α ∈ H ⊥ , where f ∈ L ( P ) . From results in [10, Chapter 3] and [36] we know that this synthesis operator L ∗ ω : L ( P ) → ℓ ( H ⊥ ) is a well-defined, bounded linear operator if, and only if, the fibers {T g p ( ω ) } p ∈ P is aBessel system. The frame operator L ∗ ω L ω of the family of fibers is called the dual Gramian andis denoted by ˜ G ω : ℓ ( H ⊥ ) → ℓ ( H ⊥ ) . Again, using results from [10, Chapter 3], the frameoperator is a bounded, linear operator acting on all of ℓ ( H ⊥ ) precisely when the fibers form aBessel system. Paying attention to the operator bounds and Bessel constants, we therefore havethe following result, extending results from [7, 8] to the case of uncountably many generators. Proposition 3.2.
Let
B > , let H ⊂ G be a closed, co-compact subgroup, and let { g p } p ∈ P ⊂ L ( G ) , where ( P, µ P ) is an admissible measure space. The following assertions are equivalent:(i) { T h g p } h ∈ H,p ∈ P is a Bessel system with bound B ,(ii) ess sup ω ∈ Ω k ˜ G ω k ≤ B ,(iii) ess sup ω ∈ Ω k L ω k ≤ √ B . In a similar fashion, it is possible to generalize [8, Proposition 4.9(2)] and the correspondingresult in [7] to the case of uncountably many generators. akobsen, Lemvig Co-compact Gabor systems on LCA groups
In the the rest of this article we will concentrate on Gabor systems. A
Gabor system in L ( G ) with generator g ∈ L ( G ) is a family of functions of the form { E γ T λ g } γ ∈ Γ ,λ ∈ Λ , where Γ ⊆ b G and Λ ⊆ G. We will usually assume that at least one of the subsets Γ ⊂ b G or Λ ⊂ G is a closed subgroup; ifeither of these subsets is not a closed subgroup, it will be assumed to be, at least, admissible asan index set (cf. the previous section). We often use that semi co-compact Gabor systems areunitarily equivalent to co-compact translation invariant systems in either time or in frequencydomain. If both Γ and Λ are closed and co-compact subgroups, we say that { E γ T λ g } γ ∈ Γ ,λ ∈ Λ is a co-compact Gabor system ; if only one of the sets Γ and Λ is a closed and co-compact subgroup,we name the Gabor system semi co-compact . If both Γ and Λ are discrete and co-compact, werecover the well-known uniform lattice Gabor systems. The fiberization technique from Theorem 3.1 will play a crucial role in the characterizationsof semi co-compact Gabor frames, presented in this subsection. From Theorem 3.1 for the TIsystem { T γ F − T λ g } γ ∈ Γ ,λ ∈ Λ , which is unitarily equivalent with { E γ T λ g } γ ∈ Γ ,λ ∈ Λ , we immediatelyhave a characterization of the frame property of Gabor systems. Proposition 4.1.
Let g ∈ L ( G ) , and let < A ≤ B < ∞ . Let Γ be a closed, co-compactsubgroup of b G , and let (Λ , Σ Λ , µ Λ ) be an admissible measure space in G . The following assertionsare equivalent:(i) { E γ T λ g } γ ∈ Γ ,λ ∈ Λ is a frame for L ( G ) with bounds A and B ,(ii) (cid:8) { g ( x + λ + α ) } α ∈ Γ ⊥ (cid:9) λ ∈ Λ is a frame for ℓ (Γ ⊥ ) with bounds A and B for a.e. x ∈ X ,where X is a Borel section of Γ ⊥ in b G . We will apply Theorem 3.1 once more to Proposition 4.1 under stronger assumptions on Λ .In the following we will always assume that Λ is a closed subgroup of G . For a moment, let useven assume that Λ = Γ ⊥ , where Γ is a closed, co-compact subgroup of b G . Note that this impliesthat Λ is discrete and countable. For uniform lattice Gabor systems the condition Λ = Γ ⊥ iscalled critical density by Gröchenig [23] since Borel sections X and Ω of the lattices Γ ⊥ and Λ ⊥ inthis case satisfy m G ( X ) m b G (Ω) = 1 . Theorem 6.5.2 in [23] states that the uniform lattice Gaborsystem { E γ T λ g } λ ∈ Λ ,γ ∈ Γ only can be frame for L ( G ) if m G ( X ) m b G (Ω) ≤ . Clearly this is not anecessary condition when either Λ or Γ is non-discrete since, for closed, co-compact subgroups,a Borel section of its annihilator has finite measure if and only if the subgroup itself is discrete.Now, back to the assumption Λ = Γ ⊥ with Γ being a (not necessarily discrete) closed, co-compact subgroup of b G . In this case, the system in Proposition 4.1(ii) is a shift invariant systemof the form { T λ ϕ x } λ ∈ Λ in ℓ (Λ) with countably many generators ϕ x := { g ( x + α ) } α ∈ Λ . We nowapply the fiberization techniques from Section 3.1 with G = Λ and H = Λ . Since the annihilator H ⊥ in this case is A ( b Λ , Λ) = { } , the fiberization map (3.1) is simply T f ( ω ) = { ˆ f ( ω ) } for ω ∈ Ω ,where Ω is a Borel section of { } in b Λ , hence, Ω = b Λ . The Fourier transform of the generator ϕ x ∈ ℓ (Λ) is ˆ ϕ x ( ω ) = X α ∈ Λ g ( x + α ) ω ( α ) , (4.1)which is the Zak transform Z Λ g ( x, ω ) of g with respect to the discrete group Λ ⊂ G . akobsen, Lemvig Co-compact Gabor systems on LCA groups By Theorem 3.1 (or a result in [7], to be more precise), { T λ ϕ x } λ ∈ Λ is a basic frame in ℓ (Λ) with bounds A and B if, and only if, { ˆ ϕ x ( ω ) } is a basic frame in ℓ ( A ( b Λ , Λ)) ∼ = C with bounds A, B for almost all ω ∈ b Λ . Now, a scalar { ˆ ϕ x ( ω ) } is a basic frame in C with bounds A and B if,and only if, its norm squared, whenever non-zero, is bounded between A and B . We concludethat { E γ T λ g } γ ∈ Λ ⊥ ,λ ∈ Λ is a Gabor basic frame in L ( G ) with bound A and B if, and only if, A ≤ (cid:12)(cid:12)(cid:12)X α ∈ Λ g ( x + α ) ω ( α ) (cid:12)(cid:12)(cid:12) ≤ B for a.e. x ∈ X, ω ∈ Ω = b Λ for which ˆ ϕ x ( ω ) = 0 . (4.2)In particular, whenever Λ = Γ ⊥ with Γ being a closed, co-compact subgroup, we see that { E γ T λ g } γ ∈ Λ ⊥ ,λ ∈ Λ is a total Gabor frame for all of L ( G ) if, and only if, A ≤ | Z Λ g ( x, ω ) | ≤ B for almost any x ∈ X, ω ∈ Ω = b Λ . Still assuming Γ = Λ ⊥ , this result can be shown to hold forany closed subgroup Λ ⊂ G [2, Theorem 2.6]. However, the next result shows a non-existencephenomenon of such continuous Gabor frames. Theorem 4.2.
Let g ∈ L ( G ) , let < A ≤ B < ∞ , and let Λ be a closed subgroup of G . Supposethat Λ is either discrete or co-compact. Then the following assertions are equivalent:(i) { E γ T λ g } γ ∈ Λ ⊥ ,λ ∈ Λ is a frame for L ( G ) with bounds A and B ,(ii) The subgroup Λ is discrete and co-compact, hence a uniform lattice, and { E γ T λ g } γ ∈ Λ ⊥ ,λ ∈ Λ is a Riesz basis for L ( G ) with bounds A and B .Proof. The implication (ii) ⇒ (i) is trivial so we only have to consider (i) ⇒ (ii).Assume first that the subgroup Λ is discrete. Then Γ = Λ ⊥ is co-compact. We use thenotation from the paragraphs preceding Theorem 4.2. Then, as shown above, assertion (i)is equivalent to { ˆ ϕ x ( ω ) } being a frame for C for almost every x ∈ X , ω ∈ b Λ . However, a oneelement set is a frame if, and only if, it is a Riesz basis with the same bounds. Now, we repeat theargument above, but in the reverse direction using a Riesz sequence variant of Theorem 3.1. By [8,Theorem 4.3] the scalar { ˆ ϕ x ( ω ) } is a Riesz basis for C if, and only if, the SI system { T λ ϕ x } λ ∈ Λ is aRiesz basis in ℓ (Λ) with the same bounds. By a result in [7], which generalizes [8, Theorem 4.3],this is equivalent to { T γ F − T λ g } γ ∈ Λ ⊥ ,λ ∈ Λ being a so-called continuous Riesz basis. However, asshown in [7] continuous
Riesz sequences only exist if Λ ⊥ is discrete. Hence, { T γ F − T λ g } γ ∈ Λ ⊥ ,λ ∈ Λ is actually a (discrete) Riesz basis. By unitarily equivalence, this implies that { E γ T λ g } γ ∈ Λ ⊥ ,λ ∈ Λ is a Riesz basis.Assume now that Λ is co-compact. Then Γ = Λ ⊥ is discrete. Note that { T λ E γ g } γ ∈ Λ ⊥ ,λ ∈ Λ isunitarily equivalent to { E γ T λ g } γ ∈ Λ ⊥ ,λ ∈ Λ and repeat the argument above for the co-compact TIsystem { T λ E γ g } γ ∈ Λ ⊥ ,λ ∈ Λ Remark . In the extreme case
Λ = G , Theorem 4.2 tell us that { T λ g } λ ∈ G cannot be a framefor L ( G ) unless G is discrete; if G is discrete, then b G is compact, and any g ∈ L ( G ) with < A ≤ | ˆ g ( ω ) | ≤ B for a.e. ω ∈ b G will generate a frame { T λ g } λ ∈ G with bounds A, B . Fordiscrete (irregular) Gabor systems in L ( R n ) such questions are studied in [12]. On the otherhand, totality in L ( G ) of the set { T λ g } λ ∈ G is achievable for both discrete and non-discrete LCAgroups G ; e.g., take any g ∈ L ( G ) with ˆ g ( ω ) = 0 for a.e. ω ∈ b G .Due to Theorem 4.2 we wish to relax the “critical” density condition Λ = Γ ⊥ , but in such away that we still can apply Zak transform methods. For regular Gabor systems (cid:8) e πiγx g ( x − λ ) : γ ∈ Γ = A Z n , λ ∈ Λ = B Z n (cid:9) (4.3)
10 of 30 akobsen, Lemvig Co-compact Gabor systems on LCA groups in L ( R n ) with A, B ∈ GL n ( R ) rational density, where A Z n ∩ B Z n is a full-rank lattice, is sucha relaxation; for n = 1 rational density simply means AB = pq ∈ Q . Our assumptions on thesubgroups Λ and Γ in the remainder of this section will mimic the setup of rational density,and the characterization will depend on a vector-valued Zak transform similar to the case of L ( R n ) [6, 40, 47].For a closed subgroup H of G the Zak transform Z H as introduced by Weil, albeit not underthis name, of a continuous function f ∈ C c ( G ) is: Z H f ( x, ω ) = Z H f ( x + h ) ω ( h ) dh for a.e. x ∈ X, ω ∈ b H. The Zak transform extends to a unitary operator from L ( G ) onto L ( G/H × b G/H ⊥ ) [2, 44].We will use the Zak transform for discrete subgroups H = Γ ⊥ , where Γ is co-compact, in whichcase, the convergence of the series Z H f ( x, α ) = P α ∈ Γ ⊥ f ( x + α ) ω ( α ) is in the L -norm for a.e. x and ω .The next result shows that the frame property of a Gabor system { E γ T λ g } λ ∈ Λ ,γ ∈ Γ in L ( G ) under certain assumptions of Λ and Γ is equivalent with the frame property of a family ofassociated Zak transformed variants of the Gabor system in C p . Theorem 4.3.
Let g ∈ L ( G ) , and let < A ≤ B < ∞ . Let Γ be a closed, co-compactsubgroup of b G . Suppose that Λ is a closed subgroup of G such that p := (cid:12)(cid:12) Γ ⊥ / (Λ ∩ Γ ⊥ ) (cid:12)(cid:12) < ∞ . Let { χ , . . . , χ p } := A ( c Γ ⊥ , Λ ∩ Γ ⊥ ) . Equip Λ with some Haar measure µ Λ , and let µ Λ / (Λ ∩ Γ ⊥ ) be theunique Haar measure on Λ / (Λ ∩ Γ ⊥ ) such that for all f ∈ L (Λ) Z Λ f ( x ) dµ Λ ( x ) = p Z Λ / (Λ ∩ Γ ⊥ ) X ℓ ∈ Λ ∩ Γ ⊥ f ( x + ℓ ) dµ Λ / (Λ ∩ Γ ⊥ ) ( ˙ x ) . Also, we let K ⊂ Λ denote a Borel section of Λ ∩ Γ ⊥ in Λ and µ K be a measure on K isometricto µ Λ / (Λ ∩ Γ ⊥ ) in the sense of (2.2) . Then the following assertions are equivalent:(i) { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a frame for L ( G ) with bounds A and B ,(ii) A k c k C p ≤ R K |h c, { Z Γ ⊥ g ( x + κ, ωχ i ) } pi =1 i C p | dµ K ( κ ) ≤ B k c k C p for all c ∈ C p , a.e. x ∈ X and ω ∈ c Γ ⊥ , where X is a Borel section of Γ ⊥ in G ,(iii) A ≤ ess inf ( x,ω ) ∈ X × c Γ ⊥ λ p ( x, ω ) , B ≥ ess sup ( x,ω ) ∈ X × c Γ ⊥ λ ( x, ω ) , where λ i ( x, ω ) denotes the i -th largest eigenvalue value of the p × p matrix ˜ G ( x, ω ) , whose ( i, j ) -th entry is ˜ G ( x, ω ) ( i,j ) = Z K Z Γ ⊥ g ( x + κ, ωχ i ) Z Γ ⊥ g ( x + κ, ωχ j ) dµ K ( κ ) . Proof.
We first remark that A ( c Γ ⊥ , Λ ∩ Γ ⊥ ) ∼ = Γ ⊥ / (Λ ∩ Γ ⊥ ) by Lemma 2.1. This shows that { χ , . . . , χ p } is well-defined due to the assumption p = (cid:12)(cid:12) Γ ⊥ / (Λ ∩ Γ ⊥ ) (cid:12)(cid:12) < ∞ .By Proposition 4.1, assertion (i) is equivalent to the sequence (cid:8) { g ( x + λ + α ) } α ∈ Γ ⊥ (cid:9) λ ∈ Λ beinga frame for ℓ (Γ ⊥ ) with bounds A and B for a.e. x ∈ X . Since Λ ∩ Γ ⊥ is a subgroup of Λ , every λ ∈ Λ can be written in a unique way as λ = µ + κ with µ ∈ Λ ∩ Γ ⊥ and κ ∈ Λ / (Λ ∩ Γ ⊥ ) . Letting
11 of 30 akobsen, Lemvig Co-compact Gabor systems on LCA groups ϕ κ := { g ( x + α + κ ) } α ∈ Γ ⊥ , we can write the above sequence as { T µ ϕ κ } µ ∈ Λ ∩ Γ ⊥ ,κ ∈ Λ / (Λ ∩ Γ ⊥ ) . Byassumption, this is a co-compact translation invariant system in ℓ (Γ ⊥ ) . The Fourier transformof ϕ κ ∈ ℓ (Γ ⊥ ) is ˆ ϕ κ ( ω ) = X α ∈ Γ ⊥ g ( x + α + κ ) ω ( α ) for a.e. ω ∈ c Γ ⊥ , hence ˆ ϕ κ ( ω ) = Z Γ ⊥ g ( x + κ, ω ) . As above, we apply the fiberization techniques from Sec-tion 3.1 with G = Γ ⊥ and H = Λ ∩ Γ ⊥ . The relationship between the measures via Weil’sformula in the assumption guarantees that the subgroups are equipped with the correct mea-sures. Since the annihilator H ⊥ in this case is A ( c Γ ⊥ , Λ ∩ Γ ⊥ ) , the fiberization map (3.1) is T f ( ω ) = { ˆ f ( ωχ ) } χ ∈ A ( c Γ ⊥ , Λ ∩ Γ ⊥ ) for ω ∈ c Γ ⊥ . By Theorem 3.1, we see that assertion (i) is equiva-lent to the system n { Z Γ ⊥ g ( x + κ, ωχ ) } χ ∈ A ( c Γ ⊥ , Λ ∩ Γ ⊥ ) o κ ∈ Λ / (Λ ∩ Γ ⊥ ) being a frame in ℓ ( A ( c Γ ⊥ , Λ ∩ Γ ⊥ )) ∼ = C p with bounds A and B for a.e. x ∈ X and ω ∈ c Γ ⊥ . Thisproves (i) ⇔ (ii).The dual Gramian matrix ˜ G ( x, ω ) is a matrix representation of the frame operator of thesystem in (ii) which shows the equivalence (ii) ⇔ (iii).Under the assumption p = (cid:12)(cid:12) Γ ⊥ / (Λ ∩ Γ ⊥ ) (cid:12)(cid:12) < ∞ , we can view { Z Γ ⊥ g ( x + κ, ωχ ) } χ ∈ A ( c Γ ⊥ , Λ ∩ Γ ⊥ ) as a column vector in C p . This vector is sometimes called a vector-valued Zak transform of g .We remark that the quotient group Λ / (Λ ∩ Γ ⊥ ) in Theorem 4.3 can be infinite, even uncountablyinfinite. If it is finite, however, we have the following simplification. Corollary 4.4.
In addition to the assertions in Theorem 4.3 assume that Λ is discrete, q := (cid:12)(cid:12) Λ / (Λ ∩ Γ ⊥ ) (cid:12)(cid:12) < ∞ and let Λ be equipped with the counting measure. Let κ i , i = 1 , . . . , q , bea set of coset representatives of Λ / (Λ ∩ Γ ⊥ ) , and let { χ , . . . , χ p } := A ( c Γ ⊥ , Λ ∩ Γ ⊥ ) . Then thefollowing assertions are equivalent.(i) { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a frame for L ( G ) with bounds A and B ,(ii) {{ Z Γ ⊥ g ( x + κ i , ωχ j ) } pj =1 } qi =1 is a frame for C p w.r.t. p − times the counting measure, i.e., A k c k C p ≤ p P qi =1 |h c, { Z Γ ⊥ g ( x + κ i , ωχ j ) } pj =1 i C p | ≤ B k c k C p for all c ∈ C p ,for a.e. x ∈ X and ω ∈ c Γ ⊥ , where X is a Borel section of Γ ⊥ in G ,(iii) A ≤ p − ess inf ( x,ω ) ∈ X × c Γ ⊥ σ p ( x, ω ) , B ≥ p − ess sup ( x,ω ) ∈ X × c Γ ⊥ σ ( x, ω ) , where σ k ( x, ω ) denotes the k -th largest singular value of the q × p matrix Φ( x, ω ) , whose ( i, j ) -th entry is Z Γ ⊥ g ( x + κ i , ωχ j ) . The matrix p − / Φ( x, ω ) is called the Zibulski-Zeevi representation; it is the transpose of thematrix representation of the synthesis operator associated with the frame in Corollary 4.4(ii).This shows that the Zibulski-Zeevi representation is possible for Gabor systems with translationalong a discrete (but not necessarily co-compact) subgroup Λ ⊂ G and modulation along aco-compact (but not necessarily discrete) subgroup Γ ⊂ b G .
12 of 30 akobsen, Lemvig Co-compact Gabor systems on LCA groups
For lattice Gabor systems (4.3) in L ( R n ) , Corollary 4.4 reduces to [6, Theorem 4.1]. Weremark that, in this case, the roles of p and q are the same as in [6, Theorem 4.1] which can beseen by an application of the second isomorphism theorem p = (cid:12)(cid:12) Γ ⊥ / (Λ ∩ Γ ⊥ ) (cid:12)(cid:12) , q = (cid:12)(cid:12) Λ / (Λ ∩ Γ ⊥ ) (cid:12)(cid:12) = (cid:12)(cid:12) (Λ + Γ ⊥ ) / Γ ⊥ (cid:12)(cid:12) , and by noting that Γ is assumed to be Z n in [6]. In particular, for regular Gabor systems in L ( R ) with time and frequency shift parameters a and b , we have ab = p/q ∈ Q , where p and q are relative prime.Using range functions, the equivalence of (i) and (ii) in all results in this subsection can beformulated for basic frames. For Corollary 4.4 this simply reads: { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a basic framein L ( G ) if, and only if, {{ Z Γ ⊥ g ( x + κ i , ωχ j ) } pj =1 } qi =1 is a basic frame in C p . In the followingExample 1 we apply this version of Corollary 4.4 to a non-discrete Gabor system and calculateits Zibulski-Zeevi representation. Example 1.
Let r ∈ N be prime. We consider Gabor systems { E γ T λ g } λ ∈ Λ ,γ ∈ Γ in L ( Z ( r ∞ )) ,where the Prüfer r -group G = Z ( r ∞ ) , the discrete group of all r n -roots of unity for all n ∈ N , isequipped with the discrete topology and multiplication as group operation. Its dual group canbe identified with the r -adic integers b G = I r . For m, n ∈ N define Λ ⊂ Z ( r ∞ ) and Γ ⊥ ⊂ Z ( r ∞ ) as all r n and r m roots of unity, respectively. Then Λ is a discrete, closed subgroup of Z ( r ∞ ) ,and Γ is a co-compact, closed subgroup of I r . Note that neither Λ nor Γ are uniform lattices.Let X and Ω denote Borel sections of the subgroups Γ ⊥ ⊂ G and Λ ⊥ ⊂ b G , respectively. For any n, m ∈ N , we have m G ( X ) m b G (Ω) = ∞ . Moreover, p = (cid:12)(cid:12) Γ ⊥ / (Λ ∩ Γ ⊥ ) (cid:12)(cid:12) = r m − min { m,n } , q = (cid:12)(cid:12) Λ / (Λ ∩ Γ ⊥ ) (cid:12)(cid:12) = r n − min { m,n } . If m ≥ n , then p = r m − n , q = 1 , and the Zibulski-Zeevi representation is (up to scaling of p − / ) given as a (row) vector of length p : Φ( x, ω ) = { Z Γ ⊥ g ( x, ωχ j ) } pj =1 , where { χ j } pj =1 = A ( c Γ ⊥ , Λ) . On the other hand, if n ≥ m , then p = 1 , q = r n − m , and theZibulski-Zeevi representation Φ( x, ω ) = { Z Γ ⊥ ( x + κ i , ω ) } qi =1 is a (column) vector of length q ,where { κ i } qi =1 is a set of coset representatives of Λ / Γ ⊥ .Thus, for any m, n ∈ N , the system { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a frame for its closed linear span, i.e.,a basic frame in L ( Z ( r ∞ )) , with bounds A and B if, and only if, A ≤ p k Φ( x, ω ) k ≤ B for almost every x ∈ X and ω ∈ c Γ ⊥ for which k Φ( x, ω ) k 6 = 0 , where Φ( x, ω ) is given as above. Remark . As an alternative to the Zak transform decomposition of g used above in part (ii) ofTheorem 4.3 and Corollary 4.4, we can use a less time-frequency symmetric variant. The detailsare as follows. By a unitary transform on C p the vector (cid:8) / √ pZ Γ ⊥ g ( x + κ, ωχ i ) (cid:9) pi =1 is mappedto the vector ψ κ ( x, ω ) := X α ∈ Λ ∩ Γ ⊥ g ( x + α + κ + ℓ i ) ω ( α ) pi =1 , (4.4)
13 of 30 akobsen, Lemvig Co-compact Gabor systems on LCA groups where ℓ i , i = 1 , . . . , p , are distinct coset representatives of Γ ⊥ / (Λ ∩ Γ ⊥ ) , and κ ∈ K . Theassertions in Theorem 4.3 are, therefore, equivalent with A k c k C p ≤ Z K |h c, ψ κ ( x, ω ) i C p | dµ K ( κ ) ≤ B k c k C p for all c ∈ C p , a.e. x ∈ X and ω ∈ c Γ ⊥ , where X is a Borel section of Γ ⊥ in G . Here µ K is the measure on K isometric to µ Λ / (Λ ∩ Γ ⊥ ) (in the sense of (2.2)) such that for all f ∈ L (Λ) Z Λ f ( x ) dµ Λ ( x ) = Z Λ / (Λ ∩ Γ ⊥ ) X ℓ ∈ Λ ∩ Γ ⊥ f ( x + ℓ ) dµ Λ / (Λ ∩ Γ ⊥ ) ( ˙ x ); note that this is different from the measure µ K used in Theorem 4.3. Then the assertions inCorollary 4.4 are equivalent to the fact that A k c k C p ≤ q X i =1 |h c, ψ κ i ( x, ω ) i C p | ≤ B k c k C p for all c ∈ C p , for a.e. x ∈ X and ω ∈ c Γ ⊥ , where X is a Borel section of Γ ⊥ in G .If we switch the assumptions on Λ and Γ and consider TI systems of the form { T λ E γ g } γ ∈ Γ ,λ ∈ Λ ,we obtain the following variant of Proposition 4.1. Proposition 4.5.
Let g ∈ L ( G ) , and let < A ≤ B < ∞ . Let Λ be a closed, co-compactsubgroup of G , and let (Γ , Σ Γ , µ Γ ) be an admissible measure space in b G . The following assertionsare equivalent:(i) { E γ T λ g } γ ∈ Γ ,λ ∈ Λ is a frame for L ( G ) with bounds A and B ,(ii) n { ˆ g ( ωγβ ) } β ∈ Λ ⊥ o γ ∈ Γ is a frame for ℓ (Λ ⊥ ) with bounds A and B for a.e. ω ∈ Ω , where Ω is a Borel section of Λ ⊥ in G . From Proposition 4.5 we get the following variant of Theorem 4.3; we leave the correspondingformulation of Corollary 4.4 to the reader.
Theorem 4.6.
Let g ∈ L ( G ) , and let < A ≤ B < ∞ . Let Λ be a closed, co-compactsubgroup of G . Suppose that Γ is a closed subgroup of b G such that p := (cid:12)(cid:12) Λ ⊥ / (Γ ∩ Λ ⊥ ) (cid:12)(cid:12) < ∞ . Let { χ , . . . , χ p } := A ( c Λ ⊥ , Γ ∩ Λ ⊥ ) . Equip Γ with some Haar measure µ Γ , and let µ Γ / (Γ ∩ Λ ⊥ ) be theunique Haar measure over Γ / (Γ ∩ Λ ⊥ ) such that for all f ∈ L (Γ) Z Γ f ( x ) dµ Γ ( x ) = p Z Γ / (Γ ∩ Λ ⊥ ) X ℓ ∈ Γ ∩ Λ ⊥ f ( x + ℓ ) dµ Γ / (Γ ∩ Λ ⊥ ) ( ˙ x ) . Also, we let K ⊂ Γ denote a Borel section of Γ ∩ Λ ⊥ in Γ and µ K be a measure on K isometricto µ Γ / (Γ ∩ Λ ⊥ ) in the sense of (2.2) . Then the following assertions are equivalent:(i) { E γ T λ g } γ ∈ Γ ,λ ∈ Λ is a frame for L ( G ) with bounds A and B ,(ii) A k c k C p ≤ R K |h c, { Z Λ ⊥ ˆ g ( ωκ, x + χ i ) } pi =1 i C p | dµ K ( κ ) ≤ B k c k C p for all c ∈ C p , a.e. ω ∈ Ω and x ∈ c Λ ⊥ , where Ω is a Borel section of Λ ⊥ in b G .
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By a result on so-called characterizing equations from [29], we now characterize when two semico-compact Gabor systems are dual frames. Using the equivalence of frame properties for sys-tems { E γ T λ g } λ ∈ Λ ,γ ∈ Γ and { T γ F − T λ g } γ ∈ Γ ,λ ∈ Λ with generator g ∈ L ( G ) yields the followingcharacterizing equations in the time domain. Theorem 4.7 ([29]) . Let Γ be a closed, co-compact subgroup of b G , and let (Λ , Σ Λ , µ Λ ) be an ad-missible measure space in G . Suppose that the two systems { E γ T λ g } γ ∈ Γ ,λ ∈ Λ and { E γ T λ h } γ ∈ Γ ,λ ∈ Λ are Bessel systems. Then the following statements are equivalent:(i) { E γ T λ g } γ ∈ Γ ,λ ∈ Λ and { E γ T λ h } γ ∈ Γ ,λ ∈ Λ are dual frames for L ( G ) ,(ii) for each α ∈ Γ ⊥ we have s α ( x ) := Z Λ g ( x − λ − α ) h ( x − λ ) dµ Λ ( λ ) = δ α, a.e. x ∈ G, (4.5)If we want to stress the dependence of the generators g and h in (4.5), we use the notation s g,h,α : G → C . Corollary 4.8.
Let Γ be a closed, co-compact subgroup of b G , and let (Λ , Σ Λ , µ Λ ) be an admissiblemeasure space in G . The family { E γ T λ g } γ ∈ Γ ,λ ∈ Λ is an A -tight frame for L ( G ) if and only if s g,g,α ( x ) = A δ α, a.e. for each α ∈ Γ ⊥ . Example 2.
Let g ∈ L ( G ) and consider { E γ T λ g } γ ∈ b G,λ ∈ Λ , where (Λ , Σ Λ , µ Λ ) be an admissiblemeasure space in G . By Corollary 4.8 we see that { E γ T λ g } γ ∈ b G,λ ∈ Λ is a Parseval frame for L ( G ) if, and only if, for a.e. x ∈ G Z Λ | g ( x − λ ) | dµ Λ ( λ ) = 1 . (4.6)If we take Λ = G with the Haar measure, then equation (4.6) becomes simply k g k = 1 which isthe well-known inversion formula for the short-time Fourier transform [23, 24].Suppose now that G contains a uniform lattice. Take Λ as a uniform lattice in G , and let X denote a (relatively compact) Borel section of Λ in G . Equation (4.6) becomes X λ ∈ Λ | g ( x − λ ) | = | X | − . Let g , . . . , g r ∈ L ( G ) be functions positive on X with support supp g i ⊂ X so that g i is constanton X for at least one index i . Following [13], the function on G defined by the r -fold convolution W r := g X ∗ g X ∗ . . . ∗ g r X is called a weighted B-spline of order r . As shown in [13], the function W r is non-negativeand satisfies a partition of unity condition up to a constant, say P λ ∈ Λ W r ( x − λ ) = C r . Take g ∈ L ( G ) so that | g ( x ) | = 1 C r | X | W r ( x ) , e.g., g ( x ) = 1( C r | X | ) / p W r ( x ) . Then { E γ T λ g } γ ∈ b G,λ ∈ Λ is a Parseval frame.Viewing Gabor systems as unitarily equivalent to { T λ E γ g } γ ∈ Γ ,λ ∈ Λ , we arrive at characterizingequations for duality in the frequency domain.
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Theorem 4.9 ([29]) . Let Λ be a closed, co-compact subgroup of G , and let (Γ , Σ Γ , µ Γ ) be an ad-missible measure space in b G . Suppose that the two systems { E γ T λ g } γ ∈ Γ ,λ ∈ Λ and { E γ T λ h } γ ∈ Γ ,λ ∈ Λ are Bessel systems. Then the following statements are equivalent:(i) { E γ T λ g } γ ∈ Γ ,λ ∈ Λ and { E γ T λ h } γ ∈ Γ ,λ ∈ Λ are dual frames for L ( G ) ,(ii) for each β ∈ Λ ⊥ we have t β ( ω ) := Z Γ ˆ g ( ωγ − β − )ˆ h ( ωγ − ) dµ Γ ( γ ) = δ β, a.e. ω ∈ b G. (4.7)As for s g,h,α we write t g,h,β : b G → C for t β in (4.7) if we want to stress the dependence of thegenerators g and h . Corollary 4.10.
Let Λ be a closed, co-compact subgroup of G , and let (Γ , Σ Γ , µ Γ ) be an admissiblemeasure space in b G . The family { E γ T λ g } γ ∈ Γ ,λ ∈ Λ is an A -tight frame for L ( G ) if and only if t g,g,β ( x ) = A δ β, a.e. for each β ∈ Λ ⊥ . Let us now consider co-compact Gabor systems, i.e., we take both Λ and Γ to be closed,co-compact subgroups. We first remark that in this case, under the Bessel system assumption,we have equivalence of conditions (4.5) and (4.7). More importantly, s g,h,α and t g,h,β can bewritten as a Fourier series. Remark . (i) For g, h ∈ L ( G ) assume that two co-compact Gabor systems { E γ T λ g } λ ∈ Λ ,γ ∈ Γ and { E γ T λ h } λ ∈ Λ ,γ ∈ Γ are Bessel systems with bounds B g and B h , respectively. By anapplication of Cauchy-Schwarz’ inequality and [29, Proposition 3.3], we see that s g,h,α ∈ L ∞ ( G ) ; to be precise: | s g,h,α ( x ) | ≤ B / g B / h for a.e. x ∈ G. (ii) Note that s g,h,α : G → C is Λ -periodic. Furthermore, G/ Λ is compact and s α is uniformlybounded, we can therefore consider s g,h,α as a function in L ( G/ Λ) and its Fourier seriesis given by s g,h,α ( x ) = X β ∈ Λ ⊥ c α,β β ( x ) with c α,β = Z G/ Λ s g,h,α ( ˙ x ) β ( ˙ x ) d ˙ x. We can compute the Fourier coefficients c α,β directly using Weil’s formula: c α,β = Z G/ Λ s g,h,α ( ˙ x ) β ( ˙ x ) d ˙ x = Z G/ Λ Z Λ g ( x − λ − α ) h ( x − λ ) β ( x − λ ) dλ d ˙ x = Z G h ( x ) β ( x ) g ( x − α ) dx = h h, E β T α g i . (4.8)(iii) Similarly, we find t g,h,β ( ω ) = P α ∈ Γ ⊥ h ˆ h, E α T β ˆ g i ω ( α ) .
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Let us begin with the definition of the frame operator. Let g ∈ L ( G ) , and let Λ ⊂ G , Γ ⊂ b G beclosed subgroups. If { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a Bessel system, the frame operator introduced in (2.6)reads: S ≡ S g,g : L ( G ) → L ( G ) , S = Z Γ Z Λ h · , E γ T λ g i E γ T λ g dλ dγ, given weakly by h Sf , f i = Z Γ Z Λ h f , E γ T λ g ih E γ T λ g, f i dλ dγ ∀ f , f ∈ L ( G ) . Similarly, for two Gabor Bessel systems generated by the functions g, h ∈ L ( G ) , we introducethe operator S g,h : L ( G ) → L ( G ) , S g,h = Z Γ Z Λ h · , E γ T λ g i E γ T λ h dλ dγ. (5.1)We follow the Gabor theory tradition, referring to this operator as a (mixed) frame operator. Ifwe want to emphasize the role of Λ and Γ , we denote this operator S g,h, Λ , Γ , where Λ specifiesthe translation subgroup and Γ the modulation subgroup.As in Gabor theory on L ( R n ) , it is straightforward to show that the frame operator commuteswith time-frequency shifts with respect to the groups Λ and Γ . Lemma 5.1.
Suppose that Γ and Λ are closed subgroups. Let g, h ∈ L ( G ) and let { E γ T λ g } λ ∈ Λ ,γ ∈ Γ , { E γ T λ h } λ ∈ Λ ,γ ∈ Γ be Bessel systems. Then, for all γ ∈ Γ and λ ∈ Λ , the following holds:(i) S g,h E γ T λ = E γ T λ S g,h ,(ii) If { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a frame, then S − E γ T λ = E γ T λ S − . Lemma 5.1 implies that the canonical dual of a Gabor frame again is a Gabor system of theform { E γ T λ h } λ ∈ Λ ,γ ∈ Γ , where h = S − g . Finally, we note that by a direct application of thePlancherel theorem, one can show that for all f , f ∈ L ( G ) , h S g,h, Λ , Γ f , f i = h S ˆ g, ˆ h, Γ , Λ ˆ f , ˆ f i , where Λ and Γ are only assumed to be measure spaces. In applications of our results, one often needs to show that the Gabor system { E γ T λ g } λ ∈ Λ ,γ ∈ Γ generated by g ∈ L ( G ) constitutes a Bessel family. This task, however, can be non-trivial,and even if g generates a Bessel system for subgroups Λ and Γ , it may not generate a Besselsystem for another pair of translation and modulation groups Λ and Γ . A solution to thisproblem is to consider functions in the Feichtinger algebra S ( G ) . It follows from [17, Theorem3.3.1] that Gabor systems { E γ T λ g } λ ∈ Λ ,γ ∈ Γ with respect to any two uniform lattices Λ and Γ in R n generated by functions in S ( R n ) are Bessel systems. The proof relies on properties of theWiener-Amalgam spaces. The purpose of this section is to give an alternate proof in the settingof LCA groups that any g ∈ S ( G ) generates a Bessel system { E γ T λ g } λ ∈ Λ ,γ ∈ Γ for any two closedsubgroups Λ ⊂ G and Γ ⊂ b G .
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Let g ∈ C c ( G ) be a non-zero function with F g ∈ L ( b G ) . The Feichtinger algebra S ( G ) isthen defined as follows: S ( G ) := (cid:8) f : G → C : f ∈ L ( G ) and Z G Z b G |V g f ( x, ω ) | dω dx < ∞ (cid:9) , where V g f ( x, ω ) := R G f ( t ) ω ( t ) g ( t − x ) dt is the short time Fourier transform of f with thewindow g . Equipped with the norm k f k S := R G × b G |V g f ( x, ω ) | dωdx , the function space S ( G ) isa Fourier-invariant Banach space that is dense in L ( G ) and whose members are continuous andintegrable functions. Moreover, S ( G ) is continuously embedded in L ( G ) , that is, there existsa constant C > such that k f k L ( G ) ≤ C k f k S ( G ) for all f ∈ S ( G ) .If g, h ∈ S ( G ) , then V g h ∈ S ( G × b G ) . Furthermore, for any closed subgroup H ⊂ G the restriction mapping R H : S ( G ) → S ( H ) , ( R H f )( x ) := f ( x ) , x ∈ H is a surjective, bounded and linear operator. We refer the reader to [16, 17, 21] for a detailedintroduction to S ( G ) .In order to prove Theorem 5.4, we need the following two results. Lemma 5.2 relies onproperties (ii) and (iv) from above, whereas Lemma 5.3 is an adaptation of [29, Lemma 2.2]. Lemma 5.2.
Let H be a closed subgroup in G and let a ∈ G , g ∈ S ( G ) . Then there exists someconstant K H > which depends on H such that Z H | g ( x − a ) | dµ H ( x ) ≤ K H k g k S ( G ) for all a ∈ G. Proof.
The result follows from the fact that S ( H ) is continuously embedded in L ( H ) and theboundedness of the restriction mapping: Z H | g ( x − a ) | dµ H ( x ) = kR H ( T a g ) k L ( H ) ≤ C kR H ( T a g ) k S ( H ) ≤ CC H k T a g k S ( G ) = CC H k g k S ( G ) . Here we also used that the S -norm is invariant under translation. Now take K H = CC H . Lemma 5.3.
Let g ∈ L ( G ) and Γ ⊂ b G be a closed subgroup. For all f ∈ C c ( G ) Z Γ |h f, E γ T λ g i| dµ Γ ( γ ) = Z G Z Γ ⊥ f ( x ) f ( x − α ) T λ g ( x ) T λ g ( x − α ) dµ Γ ⊥ ( α ) dµ G ( x ) . (5.2)With these results in hand, we can prove that functions in S ( G ) always generate GaborBessel systems. Theorem 5.4.
Let g ∈ S ( G ) and let Λ ⊂ G and Γ ⊂ b G be closed subgroups. Then { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a Bessel system with bound B = K Λ , Γ k g k S ( G ) , where K Λ , Γ is a constant that only depends on Λ and Γ .
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Proof.
From Lemma 5.3 follows that for all f ∈ C c ( G ) : Z Λ Z Γ |h f, E γ T λ g i| dγ dλ = Z Λ Z G Z Γ ⊥ f ( x ) f ( x − α ) T λ g ( x ) T λ g ( x − α ) dα dx dλ = Z Λ Z G/ Γ ⊥ Z Γ ⊥ Z Γ ⊥ g ( x − λ − α ) f ( x − α ) g ( x − λ − α ′ ) f ( x − α ′ ) dα dα ′ d ˙ x dλ. In the latter equality we used Weil’s formula and a change of variables α + α ′ α . An applicationof the triangle inequality and the Cauchy-Schwarz inequality now yields the following estimate: Z Λ Z Γ |h f, E γ T λ g i| dµ Γ ( γ ) dµ Λ ( λ ) ≤ Z G/ Γ ⊥ Z Γ ⊥ Z Γ ⊥ (cid:12)(cid:12) f ( x − α ) f ( x − α ′ ) (cid:12)(cid:12) Z Λ (cid:12)(cid:12) g ( x − λ − α ) g ( x − λ − α ′ ) (cid:12)(cid:12) dλ dα dα ′ d ˙ x ≤ Z G/ Γ ⊥ (cid:16) Z Γ ⊥ (cid:12)(cid:12) f ( x − α ) (cid:12)(cid:12) Z Λ Z Γ ⊥ (cid:12)(cid:12) g ( x − λ − α ) g ( x − λ − α ′ ) (cid:12)(cid:12) dα ′ dλ dα (cid:17) / (cid:16) Z Γ ⊥ (cid:12)(cid:12) f ( x − α ′ ) (cid:12)(cid:12) Z Λ Z Γ ⊥ (cid:12)(cid:12) g ( x − λ − α ) g ( x − λ − α ′ ) (cid:12)(cid:12) dα dλ dα ′ (cid:17) / d ˙ x. (5.3)The order of integration can be rearranged due to Tonelli’s theorem. We now apply Proposi-tion 5.2 to the two innermost integrals and find that there exists a constant K Λ , Γ > suchthat Z Λ Z Γ ⊥ (cid:12)(cid:12) g ( x − λ − α ) g ( x − λ − α ′ ) (cid:12)(cid:12) dα dλ = Z Λ | g ( x − λ − α ′ ) (cid:12)(cid:12) Z Γ ⊥ (cid:12)(cid:12) g ( x − λ − α ) | dα dλ ≤ K Λ , Γ k g k S ( G ) , where α ′ ∈ Γ ⊥ . Using this inequality in (5.3) yields the Bessel bound: Z Λ Z Γ |h f, E γ T λ g i| dµ Γ ( γ ) dµ Λ ( λ ) ≤ Z G/ Γ ⊥ (cid:16) Z Γ ⊥ (cid:12)(cid:12) f ( x − α ) (cid:12)(cid:12) K Λ , Γ k g k S ( G ) (cid:17) / (cid:16) Z Γ ⊥ (cid:12)(cid:12) f ( x − α ′ ) (cid:12)(cid:12) K Λ , Γ k g k S ( G ) (cid:17) / dµ G/ Γ ⊥ ( ˙ x )= K Λ , Γ k g k S ( G ) k f k L ( G ) . Since C c ( G ) is dense in L ( G ) , the result follows. The continuous Gabor frame operator associated with semi co-compact Gabor systems definedin (5.1) can be converted into a discrete transform called the Walnut representation. The Walnutrepresentation plays an important role the usual discrete (lattice) theory of Gabor analysis. ForGabor theory on L ( R ) the result goes back to [43] and is also presented in [24]. See [9] for adetailed analysis of the convergence properties of the Walnut representation in L ( R ) .In order to state our version of the Walnut representation, we need to introduce two densesubspaces of L ( G ) : D s := (cid:8) f ∈ L ( G ) : f ∈ L ∞ ( G ) and supp f is compact in G (cid:9) (5.4)
19 of 30 akobsen, Lemvig Co-compact Gabor systems on LCA groups and D t := (cid:8) f ∈ L ( G ) : ˆ f ∈ L ∞ ( b G ) and supp ˆ f is compact in b G (cid:9) . (5.5)Recall also the definition of s α and t β from (4.5) and (4.7), respectively. Theorem 5.5.
Let g, h ∈ L ( G ) . Let Γ be a closed, co-compact subgroup of b G , and let (Λ , Σ Λ , µ Λ ) be an admissible measure space in G . Suppose that { E γ T λ g } λ ∈ Λ ,γ ∈ Γ and { E γ T λ h } λ ∈ Λ ,γ ∈ Γ areBessel systems, and let S g,h be the associated mixed frame operator. Then S g,h f = X α ∈ Γ ⊥ M s α T α f for all f ∈ D s , (5.6) with unconditional, norm convergence in L ( G ) .Proof. By the proof of the main result in [29], we have that for all f , f ∈ D s , h S g,h f , f i = Z Λ Z Γ h f , E γ T λ g ih E γ T λ h, f i dγ dλ = X α ∈ Γ ⊥ (cid:10) M s α T α f , f (cid:11) . Moreover, the convergence is absolute and thus unconditionally. Because D s is dense in L ( G ) spaces we have that h S g,h f , f i = P α ∈ Γ ⊥ (cid:10) M s α T α f , f (cid:11) holds for all f ∈ L ( G ) . By the Orlicz-Pettis Theorem (see, e.g., [15]), this implies unconditional L -norm convergence for (5.6). Remark . If we assume g, h ∈ S ( G ) , then (5.6) extends to all of L ( G ) . Remark . In Theorem 5.5, if we instead assume that Λ is a closed, co-compact subgroup of G and that (Γ , Σ Γ , µ Γ ) is an admissible measure space in b G , then F S g,h f = X β ∈ Λ ⊥ M t β T β ˆ f for all f ∈ D t (5.7)holds.We can now easily show the following result. Corollary 5.6. (i) Under the assumptions of Theorem 5.5 and if { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a framewith bounds A and B , then A ≤ Z Λ | g ( x + λ ) | dλ ≤ B a.e. x ∈ G. (ii) Under the assumptions of Remark 6 and if { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a frame with bounds A and B , then A ≤ Z Γ | ˆ g ( ωγ ) | dγ ≤ B a.e. ω ∈ b G. In either case, if { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a Bessel system with bound B , then the upper bound holds.
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Proof. If { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a frame, then, in particular, A k f k ≤ h S g,g f, f i ≤ B k f k for all f ∈ D s ( G ) . Pick now a function f ∈ D s ( G ) so that the support of f lies within a fundamental domain of thediscrete group Γ ⊥ ⊂ G . Then, by (5.6), A k f k ≤ h X α ∈ Γ ⊥ M s α T α f, f i ≤ B k f k ⇔ A k f k ≤ h s f, f i ≤ B k f k ⇔ A Z G | f ( x ) | dx ≤ Z G (cid:16) Z Λ | g ( x + λ ) | dλ (cid:17) | f ( x ) | dx ≤ B Z G | f ( x ) | dx. From this assertion (i) follows. By use of (5.7), one proves assertion (ii) in the same fashion.
The Walnut representation was formulated for semi co-compact Gabor systems. In case both Λ and Γ are co-compact, closed subgroups, we can offer a more time-frequency symmetricalrepresentation of the Gabor frame operator; this is the so-called Janssen representation. Theorem 5.7.
Let g, h ∈ L ( G ) and let Λ ⊂ G, Γ ⊂ b G be closed, co-compact subgroups such that { E γ T λ g } λ ∈ Λ ,γ ∈ Γ and { E γ T λ h } λ ∈ Λ ,γ ∈ Γ are Bessel systems. Suppose that the pair ( g, h ) satisfies condition A : X α ∈ Γ ⊥ X β ∈ Λ ⊥ (cid:12)(cid:12) h h, E β T α g i (cid:12)(cid:12) < ∞ . (5.8) Then S g,h = X α ∈ Γ ⊥ X β ∈ Λ ⊥ h h, E β T α g i E β T α (5.9) with absolute convergence in the operator norm.Proof. Define the operator ˜ S : L ( G ) → L ( G ) by ˜ S = X α ∈ Γ ⊥ X β ∈ Λ ⊥ h h, E β T α g i E β T α . This series converges absolutely in the operator norm by (5.8). Hence, the convergence is uncon-ditionally. Replacing s α in the Walnut representation by its Fourier series representation fromRemark 4 yields h S g,h f , f i = h X α ∈ Γ ⊥ M s α T α f , f i = h X α ∈ Γ ⊥ X β ∈ Λ ⊥ h h, E β T α g i β ( x ) T α f , f i = X α ∈ Γ ⊥ X β ∈ Λ ⊥ h h, E β T α g ih E β T α f , f i = h ˜ Sf , f i for f , f ∈ D s . Since D s is dense in L ( G ) , it follows that S g,h = ˜ S .
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Note that (5.9) indicates convergence in the uniform operator topology, while Walnut’s rep-resentation, on the other hand, conveyed convergence in the strong operator topology.For generators g, h ∈ S ( G ) in Feichtinger’s algebra, the assumptions of the Janssen represen-tation in Theorem 5.7 are automatically satisfied. The Bessel condition follows from Theorem 5.4,while (5.8) follows from the next result. Proposition 5.8.
Let g, h ∈ S ( G ) , and let Λ and Γ be closed subgroups in G and b G , respectively.The pair ( g, h ) satisfies (5.8) , that is, Z Λ ⊥ Z Γ ⊥ |h g, E β T α h i| dα dβ < ∞ . Proof.
By [17, Corollary 7.6.6] we have that g, h ∈ S ( G ) implies ( x, ω )
7→ h g, E ω T x h i ∈ S ( G × b G ) . If we restrict this mapping to Γ ⊥ × Λ ⊥ ⊂ G × b G and use that S is continuously embeddedinto L , we find that (5.8) is satisfied.The next version of the Janssen representation holds for arbitrary (not necessarily co-compact)closed subgroups Λ ⊂ G, Γ ⊂ b G . It is called the fundamental identity of Gabor analysis (FIGA).In [19] Feichtinger and Luef give a detailed answer to when (5.10) holds in the setting of R n , seealso [17, 21] for related results. The FIGA was first proved by Rieffel [38] for generators g, h inthe Schwartz-Bruhat space S ( G ) . Rieffel’s proof uses the Poisson summation formula and alsoholds for the non-separable case with closed subgroups in G × b G ; it is also possbile to give anargument based on Janssen’s proof for (lattice) Gabor systems in L ( R ) [30, 31]. Theorem 5.9.
Let f , f , g, h ∈ L ( G ) , and let Λ ⊂ G, Γ ⊂ b G be closed subgroups. Assume that { E γ T λ g } λ ∈ Λ ,γ ∈ Γ and { E γ T λ h } λ ∈ Λ ,γ ∈ Γ are Bessel systems. If ( α, β )
7→ h E β T α f , f ih h, E β T α g i ∈ L (Γ ⊥ × Λ ⊥ ) , then h S g,h f , f i = Z Γ ⊥ Z Λ ⊥ h h, E β T α g ih E β T α f , f i dβ dα. (5.10) The Janssen representation shows that the frame operator of a co-compact Gabor system canbe written in terms of the system { E β T α g } α ∈ Γ ⊥ ,β ∈ Γ ⊥ . In this section we present further re-sults that connect a co-compact Gabor system { E γ T λ g } λ ∈ Λ ,γ ∈ Γ with its adjoint Gabor system { E β T α g } α ∈ Γ ⊥ ,β ∈ Γ ⊥ .The time-frequency shifts in a Gabor system and its adjoint system are characterized by thefact that they commute [21, Section 3.5.3], [24, Lemma 7.4.1]. That is, for ( λ, γ ) ∈ Λ × Γ thepoint ( α, β ) ⊂ G × b G belongs to Γ ⊥ × Λ ⊥ if and only if ( E γ T λ )( E β T α ) = ( E β T α )( E γ T λ ) . We remind the reader our convention equipping the annihilator of Λ and Γ with the countingmeasure. The following results will, therefore, only after appropriate modification take thefamiliar form of the lattice Gabor theory in, e.g., L ( R n ) .
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Bessel bound duality states that a co-compact Gabor system is a Bessel system with bound B if,and only if, the discrete adjoint Gabor system { E β T α g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ is a Bessel system with bound B . The result is stated in Proposition 6.4, and its proof is divided into two parts, Lemma 6.2and 6.3.We begin with the definition of the operator L x : D ( L x ) → L (Λ) with D ( L x ) ⊂ ℓ (Γ ⊥ ) . Let x ∈ G , let g ∈ L ( G ) be given and let { c α } α ∈ Γ ⊥ be a finite sequence. Then for almost every x ∈ G we define the linear operator L x ( { c α } α ∈ Γ ⊥ ) = λ X α ∈ Γ ⊥ g ( x − λ − α ) c α , D ( L x ) = c (Γ ⊥ ) . (6.1)Note that L x essentially (up to complex conjugations, etc.) is the analysis operator, as introducedin (3.2), of the family of fibers associated with the TI system (cid:8) T γ F − T λ g (cid:9) γ ∈ Γ ,λ ∈ Λ . In light ofProposition 3.2, we therefore have the following result. Lemma 6.1. If { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a Bessel system with bound B , then for almost every x ∈ G the operator L x extends to a linear, bounded operator with domain ℓ (Γ ⊥ ) and bound B / . Let us now show one direction of the Bessel duality between a co-compact Gabor system andits adjoint.
Lemma 6.2.
Let Λ ⊂ G and Γ ⊂ b G be closed, co-compact subgroups. If { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is aBessel system with bound B , then { E β T α g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ is a Bessel system with bound B .Proof. We consider the discrete Gabor system { E β T α g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ and its associated synthesismapping F : ℓ (Γ ⊥ × Λ ⊥ ) → L ( G ) , F c ( α, β ) = X α ∈ Γ ⊥ X β ∈ Λ ⊥ c ( α, β ) E β T α g. We will show that F is a well-defined, linear and bounded operator with k F k ≤ B / ; the resultthen follows from [10, Theorem 3.2.3]. To this end, let c ∈ ℓ (Γ ⊥ × Λ ⊥ ) be a finite sequence andfor each x ∈ G consider m α ( x ) := X β ∈ Λ ⊥ c ( α, β ) β ( x ) , α ∈ Γ ⊥ . (6.2)It is clear that { m α ( x ) } α ∈ Γ ⊥ is a finite sequence as well. Note that m α as a function of x ∈ G is constant on cosets of Λ . Thus m α defines a function on G/ Λ , which we will denote by m α ( ˙ x ) .By use of the identification G/ Λ ∼ = c Λ ⊥ and the Parseval equality, we find Z G/ Λ | m α ( ˙ x ) | dµ G/ Λ ( ˙ x ) = Z c Λ ⊥ (cid:12)(cid:12)(cid:12) X β ∈ Λ ⊥ c ( α, β ) β ( x ) (cid:12)(cid:12)(cid:12) dµ c Λ ⊥ ( x )= k c ( α, β ) k ℓ (Λ ⊥ ) = X β ∈ Λ ⊥ | c ( α, β ) | . (6.3)By definition we have thatF c = X α ∈ Γ ⊥ X β ∈ Λ ⊥ c ( α, β ) E β T α g = X α ∈ Γ ⊥ M m α T α g.
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Using this expression together with Weil’s formula we find the following for the norm of F c : k F c k = Z G | F c ( x ) | dµ G ( x ) = Z G X α,α ′ ∈ Γ ⊥ m α ( x ) g ( x − α ) m α ′ ( x ) g ( x − α ′ ) dµ G ( x )= Z G/ Λ Z Λ (cid:16) X α ∈ Γ ⊥ m α ( ˙ x ) g ( x − λ − α ) (cid:17)(cid:16) X α ′ ∈ Γ ⊥ m α ′ ( ˙ x ) g ( x − λ − α ′ ) (cid:17) dµ Λ ( λ ) dµ G/ Λ ( ˙ x )= Z G/ Λ k L x m α ( ˙ x ) k L (Λ) dµ G/ Λ ( ˙ x ) . (6.4)The rearranging of the summation is possible because the summations over Γ ⊥ are finite. Since { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a Bessel system with bound B , we know by Lemma 6.1 that L x is boundedby B / . We therefore have that k L x m α ( ˙ x ) k ≤ B k m α ( ˙ x ) k = B X α ∈ Γ ⊥ | m α ( ˙ x ) | . Using this together with (6.3) and (6.4) yields the following inequality. k F c k ≤ B Z G/ Λ X α ∈ Γ ⊥ | m α ( x ) | dµ G/ Λ ( ˙ x ) = B X α ∈ Γ ⊥ X β ∈ Λ ⊥ | c ( α, β ) | = B k c k ℓ (Γ ⊥ × Λ ⊥ ) . We conclude that F is bounded by B / and so { E β T α g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ is a Bessel system with bound B . Note that in the classical discrete and co-compact setting we simply apply Lemma 6.2 tothe adjoint Gabor system, as it would also be discrete and co-compact. However, in our casethe Gabor system { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is co-compact and the adjoint system is discrete (and notnecessarily co-compact). We thus need another result for the reverse direction.In order to prove the reverse direction, Lemma 6.3, we will reuse calculations from Lemma6.2. Furthermore, the proof also relies on Lemma 5.3. Adapted to co-compact Γ ⊂ b G it statesthat for all f ∈ C c ( G ) Z Γ |h f, E γ T λ g i| dµ Γ ( γ ) = Z G X α ∈ Γ ⊥ f ( x ) f ( x − α ) T λ g ( x ) T λ g ( x − α ) dµ G ( x ) . (6.5) Lemma 6.3.
Let Λ ⊂ G and Γ ⊂ b G be closed, co-compact subgroups. If { E β T α g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ isa Bessel system with bound B , then { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a Bessel system with bound B .Proof. Note that for finite sequences c ∈ ℓ (Γ ⊥ × Λ ⊥ ) the calculations in (6.4) still hold. We let m α ( x ) be given as in (6.2). By assumption we know that the synthesis mapping F of the adjointGabor system { E β T α g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ is bounded by B / . We therefore have that k F c k = Z G/ Λ k L x m α ( ˙ x ) k L (Λ) dµ G/ Λ ( ˙ x ) ≤ B k c k ℓ (Γ ⊥ × Λ ⊥ ) ∀ c ∈ ℓ (Γ ⊥ × Λ ⊥ ) . By use of (6.3) we rewrite the norm of c and find Z G/ Λ k L x m α ( ˙ x ) k L (Λ) dµ G/ Λ ( ˙ x ) ≤ B Z G/ Λ k m α ( ˙ x ) k ℓ (Γ ⊥ ) dµ G/ Λ ( ˙ x ) . (6.6)
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This implies that k L x m α ( ˙ x ) k L (Λ) ≤ B k m α ( ˙ x ) k ℓ (Γ ⊥ ) . (6.7)If c ( α, β ) = 0 for all β = 1 , then m α ( x ) = c ( α, . Therefore the mapping from all finite c ∈ ℓ (Γ ⊥ × Λ ⊥ ) to m α ( x ) in (6.2) is a surjection onto all finite sequences indexed by Γ ⊥ . From(6.7) we can therefore conclude that L x is a bounded operator from all finite sequences to L (Λ) with k L x k ≤ B / . Since L x is also linear, it uniquely extends to a bounded operator from all of ℓ (Γ ⊥ ) to L (Λ) .Let now f ∈ C c ( G ) and consider the finite sequence c = { f ( x − α ) } α ∈ Γ ⊥ . Replacing { m α ( ˙ x ) } α ∈ Γ ⊥ with c in (6.6) yields the following inequality: Z G/ Γ ⊥ k L x c k L (Λ) dµ G/ Γ ⊥ ( ˙ x ) ≤ B Z G/ Γ ⊥ X α ∈ Γ ⊥ | f ( x − α ) | dµ G/ Γ ⊥ ( ˙ x ) = B k f k L ( G ) . (6.8)Concerning the left hand side of (6.8), we find that Z G/ Γ ⊥ k L x c k L (Λ) dµ G/ Γ ⊥ ( ˙ x )= Z G/ Γ ⊥ Z Λ X α,α ′ ∈ Γ ⊥ g ( x − λ − α ) f ( x − α ) g ( x − λ − α ′ ) f ( x − α ′ ) dλ dµ G/ Γ ⊥ = Z G/ Γ ⊥ Z Λ X α,α ′ ∈ Γ ⊥ g ( x − λ − α ′ − α ) f ( x − α ′ − α ) g ( x − λ − α ′ ) f ( x − α ′ ) dλ dµ G/ Γ ⊥ = Z G Z Λ X α ∈ Γ ⊥ g ( x − λ − α ) f ( x − α ) g ( x − λ ) f ( x ) dλ dµ G ( x )= Z Λ Z Γ |h f, E γ T λ g i| dµ Γ ( γ ) dµ Λ ( λ ) . (6.9)The last equality follows by (6.5). From (6.8) and (6.9) we conclude that Z Λ Z Γ |h f, E γ T λ g i| dµ Γ ( γ ) dµ Λ ( λ ) ≤ B k f k for all f ∈ C c ( G ) . Since this holds for all f in a dense subset of L ( G ) we draw the conclusion that { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a Bessel system with bound B .The combination of Lemma 6.2 and 6.3 yields the Bessel bound duality between a co-compactGabor system and its discrete adjoint system. Proposition 6.4.
Let
B > and g, h ∈ L ( G ) be given. Let Γ ⊂ G and Λ ⊂ b G be closed,co-compact subgroups. Then { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a Bessel system with bound B if, and only if, { E β T α g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ is a Bessel system with bound B . We now turn our attention to a characterization of dual co-compact Gabor frame generators by abiorthogonality condition of the corresponding (discrete) adjoint Gabor systems. Feichtinger andKozek [17] proved the Wexler-Raz biorthogonality relations for Gabor systems with translationand modulation along uniform lattices on elementary LCA groups, i.e., G = R n × T ℓ × Z k × F m ,where F m is a finite group. For a proof in the discrete and finite setting and on the real line werefer to the original papers [45] and [31].
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Theorem 6.5.
Let Λ ⊂ G and Γ ⊂ b G be closed, co-compact subgroups. Let g, h ∈ L ( G ) and assume that { E γ T λ g } λ ∈ Λ ,γ ∈ Γ and { E γ T λ h } λ ∈ Λ ,γ ∈ Γ are Bessel systems. Then the two Gaborsystems are dual frames if, and only if, h h, E β T α g i = δ β, δ α, ∀ α ∈ Γ ⊥ , β ∈ Λ ⊥ . (6.10) Proof.
Assume that the two Gabor systems are dual frames. Then, for each α ∈ Γ ⊥ , we have s α = δ α, for a.e. x ∈ G . By uniqueness of the Fourier coefficients (4.8), the conclusion in (6.10)follows. The converse direction is immediate. Remark . (i).(i) From equation (6.10) with α ′ ∈ Γ ⊥ , β ′ ∈ Λ ⊥ we find δ β, δ α, = h h, E β T α g i = h E β ′ T α ′ h, β ( α ) E β ′ β T α ′ + α g i . And thus the Wexler-Raz biorthogonality relations (6.10) can equivalently be stated as h E β T α h, E β ′ T α ′ g i = δ α,α ′ δ β,β ′ ∀ α, α ′ ∈ Γ ⊥ , β, β ′ ∈ Λ ⊥ . (ii) For canonical dual frames { E γ T λ g } λ ∈ Λ ,γ ∈ Γ and (cid:8) E γ T λ S − g (cid:9) λ ∈ Λ ,γ ∈ Γ , the biorthogonal se-quences { E β T α g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ and (cid:8) E β T α S − g (cid:9) α ∈ Γ ⊥ ,β ∈ Λ ⊥ are actually dual Riesz bases forthe subspace span { E β T α g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ , see [31, Proposition 3.3]. The duality principle for lattice Gabor systems in L ( R n ) was proven simultaneously by threegroups of authors, Daubechies, Landau and Landau [14], Janssen [31], and Ron and Shen [40].Theorem 6.7 below generalizes this principle to co-compact Gabor systems in L ( G ) . Our proofof the duality principle relies on the following result on Riesz sequences in abstract Hilbert spaces,cf. Definition 2.3. It is a subspace variant of [11, Theorem 3.4.4] and [25, Theorem 7.13]; itsproof is due to Ole Christensen. Theorem 6.6.
Let { f k } be a sequence in a Hilbert space. Then the following statements areequivalent:(a) { f k } is a Riesz sequence with lower bound A and upper bound B ,(b) { f k } is a Bessel system with bound B and possesses a biorthogonal system { g k } that is alsoa Bessel system with bound A − .Proof. Assume that (a) holds. Set V = span { f k } . Let { g k } be the unique dual Riesz sequenceof { f k } in V so that span { g k } = V . This implies (b).Assume that (b) holds. Since { f k } and { g k } are biorthogonal, it follows that f j = X k h f j , g k i f k for all j . By linearity, we have, for any f ∈ span { f k } , f = X k h f, g k i f k .
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This formula extends to span { f k } by continuity. Now, for any f ∈ span { f k } , we have k f k = |h f, f i| = (cid:12)(cid:12)(cid:12)X k h f, g k ih f k , f i (cid:12)(cid:12)(cid:12) ≤ (cid:18)X k |h f, g k i| X k |h f, f k i| (cid:19) / ≤ A − / k f k (cid:18)X k |h f, f k i| (cid:19) / . (6.11)We see that { f k } is a frame sequence with lower frame bound A ; by assumption the upper framebound is B . By the fact that { f k } possesses a biorthogonal sequence, it follows that { f k } is, infact, a Riesz sequence with the same bounds. Theorem 6.7.
Let g ∈ L ( G ) . Let Λ ⊂ G and Γ ⊂ b G be closed, co-compact subgroups. Then { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a frame for L ( G ) with bounds A and B if, and only if, { E β T α g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ Riesz sequence with bounds A and B .Proof. Let { E γ T λ g } λ ∈ Λ ,γ ∈ Γ be a frame with bounds A and B . The canonical dual frame (cid:8) E γ T λ S − g (cid:9) λ ∈ Λ ,γ ∈ Γ has bounds B − and A − . By Proposition 6.4, the sequences { E β T α g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ and { E β T α S − g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ are Bessel systems with bound B and A − , respectively. By Wexler-Raz biorthogonal relations, these two families are biorthogonal, hence, by Theorem 6.6, { E β T α g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ is a Riesz sequence with bounds A and B .Conversely, suppose { E β T α g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ is a Riesz sequence with bounds A and B . The dualRiesz sequence of { E β T α g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ is of the form { E β T α h } α ∈ Γ ⊥ ,β ∈ Λ ⊥ for some h ∈ L ( G ) andhas bounds B − and A − . Using Proposition 6.4 we see that { E γ T λ g } λ ∈ Λ ,γ ∈ Γ has Bessel bound B . On the other hand, { E γ T λ h } λ ∈ Λ ,γ ∈ Γ has Bessel bound A − . By Wexler-Raz biorthogonalrelations, { E γ T λ g } λ ∈ Λ ,γ ∈ Γ and { E γ T λ h } λ ∈ Λ ,γ ∈ Γ are dual frames. By a computation as in (6.11),we see that A is a lower frame bound for { E γ T λ g } λ ∈ Λ ,γ ∈ Γ .The co-compactness assumption on Λ and Γ is a natural framework for the duality principle.Indeed, if the Gabor system is not co-compact, the adjoint system is not discrete. However, weknow by a result of Bownik and Ross [7] that continuous Riesz sequences do not exist. Hence, ifeither Λ or Γ is not co-compact, the adjoint Gabor system cannot be a Riesz “sequence”.Since a Riesz sequence with bounds A = B is an orthogonal sequence, we have the followingcorollary of Theorem 6.7. Corollary 6.8.
Let Γ and Λ be closed, co-compact subgroups. A Gabor system { E γ T λ g } λ ∈ Λ ,γ ∈ Γ is a tight frame if, and only if, { E β T α g } α ∈ Γ ⊥ ,β ∈ Λ ⊥ is an orthogonal system. In these cases, theframe bound is given by A = k g k . We end this paper with the following general remark:
Remark . We have stated the results of the current paper for Gabor systems generated by asingle function, however, most of the results can be stated for finitely or even infinitely manygenerators; the non-existence result, Theorem 4.2, is of course an exception to this rule.
Acknowledgments.
The authors thank Ole Christensen for useful discussions and for theproof of Theorem 6.6. The first-named author also thanks Hans G. Feichtinger for discussionsand pointing out references concerning S .
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