Gabor frame characterisations of generalised modulation spaces
aa r X i v : . [ m a t h . F A ] F e b GABOR FRAME CHARACTERISATIONS OF GENERALISEDMODULATION SPACES
ANDREAS DEBROUWERE AND BOJAN PRANGOSKI
Abstract.
We obtain Gabor frame characterisations of modulation spaces definedvia a class of translation-modulation invariant Banach spaces of distributions that wasrecently introduced in [10]. We show that these spaces admit an atomic decompositionthrough Gabor expansions and that they are characterised by summability propertiesof their Gabor coefficients. Furthermore, we construct a large space of admissiblewindows. This generalises several fundamental results for the classical modulationspaces M p,qw . Due to the absence of solidity assumptions on the Banach spaces definingthese modulation spaces, the methods used for the spaces M p,qw (or, more generally,in coorbit space theory) fail in our setting and we develop here a new approach basedon the twisted convolution. Introduction
Modulation spaces, introduced by Feichtinger [12] in 1983, are one of the princi-pal objects in time-frequency analysis. Their properties were thoroughly studied byFeichtinger and Gr¨ochenig [12, 13, 14, 15, 18], often in the more general setting ofcoorbit spaces. Nowadays, they are widely accepted as an indispensable tool in variousbranches of analysis; see e.g. [1, 6, 7, 26]. A key feature of the modulation spaces isthat they can be described in a discrete fashion via Gabor frames. Apart from theirinherent significance, such characterisations have also turned out to be very useful inapplications, e.g., in the study of pseudo-differential operators; see [7] and the refer-ences therein. We refer to the monograph [19] for an account of results and applicationsof modulation spaces.Usually, modulation spaces are defined via weighted mixed-norm spaces [19] or,more generally – in the context of coorbit spaces – via translation invariant solidBanach function spaces [13]. In [10] a new class of Banach spaces was proposed todefine modulation spaces: A Banach space F is said to be a translation-modulationinvariant Banach space of (tempered) distributions (TMIB) on R n if F satisfies thedense continuous inclusions S ( R n ) ֒ → F ֒ → S ′ ( R n ), F is translation and modulationinvariant, and the operator norms of the translation and modulation operators on F are polynomially bounded. We refer to [10] (see also [9]) for a systematic study ofTMIB and their duals (called DTMIB). It is important to point out that TMIB arenot necessarily solid (in the sense of [13]). A natural example of a non-solid TMIB is Mathematics Subject Classification. Primary.
Secondary.
Key words and phrases.
Gabor frames; modulation spaces; translation-modulation invariant Ba-nach spaces of distributions; amalgam spaces.A. Debrouwere was supported by FWO-Vlaanderen through the postdoctoral grant 12T0519N. given by L p b ⊗ π L p , 1 < p ≤ M F associatedto a TMIB or DTMIB F on R n consists of all those tempered distributions on R n whose short-time Fourier transform belongs to F . The basic properties of these spaceswere established in [10, Section 4].A natural question that arises is whether modulation spaces defined via TMIB andDTMIB may be described in terms of Gabor frames. The goal of the present paper isto give an affirmative answer to this question, namely, we show that these spaces admitan atomic decomposition through Gabor expansions and that they are characterised bysummability properties of their Gabor coefficients. The significance of this lies in thefact that the modulation spaces give a scale of measurement of the regularity and decayproperties of tempered distributions and such characterisations allow these propertiesto be quantified in a discrete way by means of Gabor coefficients.We now describe the content of this paper in some more detail and point out themain difference between our setting and the classical one involving solid spaces. For( x, ξ ) ∈ R n , we write π ( x, ξ ) = M ξ T x , where T x f ( t ) = f ( t − x ) and M ξ f ( t ) = f ( t ) e πit · ξ denote the translation and modulation operators on R n . We also set ˇ f ( t ) = f ( − t ). Fixa lattice Λ in R n and a bounded open neighbourhood U of the origin in R n such thatthe family of sets { λ + U | λ ∈ Λ } is pairwise disjoint. Given a solid TMIB F on R n ,we associate to it the following discrete solid Banach space on Λ [13, Definition 3.4] F d (Λ) = ( c = ( c λ ) λ ∈ Λ ∈ C Λ (cid:12)(cid:12)(cid:12) X λ ∈ Λ c λ λ + U ∈ F ) , where 1 λ + U is the characteristic function of the set λ + U , with norm k c k F d (Λ) = k P λ ∈ Λ | c λ | λ + U k F . The modulation space M F admits the following characterisationin terms of Gabor frames [13, 18] (see [19, Section 12.2] for the classical modulationspaces M p,qw = M L p,qw ). Theorem 1.1.
Let F be a solid TMIB on R n . Set ω F ( x, ξ ) = k T ( x,ξ ) k L ( F ) , ( x, ξ ) ∈ R n . Let ψ, γ ∈ M , { ω F , ˇ ω F } . Then, the analysis operator C ψ : M F → F d (Λ) , f ( V ψ f ( λ )) λ ∈ Λ and the synthesis operator D γ : F d (Λ) → M F , ( c λ ) λ ∈ Λ X λ ∈ Λ c λ π ( λ ) γ are well-defined and continuous, and the series P λ ∈ Λ c λ π ( λ ) γ is unconditionally con-vergent in F for each c ∈ F d (Λ) . If in addition ( ψ, γ ) is a pair of dual windows on Λ ,then there are A, B > such that A k f k M F ≤ k ( V ψ f ( λ )) λ ∈ Λ k F d (Λ) ≤ B k f k M F , f ∈ M F , and the following expansions hold f = X λ ∈ Λ V ψ f ( λ ) π ( λ ) γ = X λ ∈ Λ V γ f ( λ ) π ( λ ) ψ, f ∈ M F , ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 3 where both series are unconditionally convergent in F . In fact, Theorem 1.1 holds for more irregular samplings sets than lattices [13, 18].The standard proof of Theorem 1.1 [13, 18, 19] is based on the following two funda-mental properties of the STFT:(1.1) | V ψ ( π ( x, ξ ) f ) | = | T ( x,ξ ) V ψ f | and | V ψ f | ≤ | ( γ, ψ ) L | | V ψ f | ∗ | V ψ γ | , where f, ψ, γ ∈ L ( R n ) with ( γ, ψ ) L = 0; (1.1) may be extended to other spaces.Hence, Theorem 1.1 essentially reduces to prove that the mappings F → F d (Λ) , G ( G ∗ Φ( λ )) λ ∈ Λ and F d (Λ) → F, ( c λ ) λ ∈ Λ X λ ∈ Λ c λ T λ Ψare well-defined and continuous, and that the series P λ ∈ Λ c λ T λ Ψ is unconditionallyconvergent in F for each c ∈ F d (Λ), where Φ , Ψ belong to suitable function spaces on R n .Our aim is to extend Theorem 1.1 to general TMIB. However, the properties (1.1)are no longer applicable in this setting. The basic idea to overcome this problem is toview the STFT on L ( R n ) as the voice transform of the projective representation [4, 5] π : ( R n , +) → L ( L ( R n )) . The twisted translation and the twisted convolution associated to π are given by T σ ( x,ξ ) f ( t, η ) = f ( t − x, η − ξ ) e − πix · ( η − ξ ) and f g ( t, η ) = Z Z R n f ( x, ξ ) g ( t − x, η − ξ ) e − πix · ( η − ξ ) d x d ξ. Then,(1.2) V ψ ( π ( x, ξ ) f ) = T σ ( x,ξ ) V ψ f and V ψ f = 1( γ, ψ ) L V ψ f V ψ γ, where f, ψ, γ ∈ L ( R n ) with ( γ, ψ ) L = 0; (1.2) may be extended to other spaces.From this point of view, it seems natural to define the discrete space associated toa TMIB F via the twisted translation T σ , i.e., F σd (Λ) = ( c ∈ C Λ (cid:12)(cid:12)(cid:12) X λ ∈ Λ c λ T σλ χ ∈ F ) , where χ ∈ D ( U ) \{ } , with norm k c k F Bd (Λ) = k P λ ∈ Λ c λ T σλ χ k F . Then, F σd (Λ) is aBanach space that is independent of χ ∈ D ( U ) \{ } (Theorem 5.2). Moreover, F d (Λ) = F σd (Λ) if F is solid. We shall determine the discrete space associated to various TMIBfor lattices Λ = Λ × Λ , where Λ and Λ are lattices in R n (Subsection 5.4). Mostnotably, ( L b ⊗ π L ) σd (Λ × Λ ) = ℓ (Λ ; ℓ (Λ )) , (1.3) ( L b ⊗ ǫ L ) σd (Λ × Λ ) = c (Λ ; ℓ (Λ )) . A. DEBROUWERE AND B. PRANGOSKI
The main results of this paper (Theorem 6.6 and Corollary 6.7) show that Theorem1.1 holds for general TMIB F provided that F d (Λ) is replaced by F σd (Λ), the function ω F defining the admissible window class is changed to σ F , where σ F ( x, ξ ) = k T ( x,ξ ) k L ( F ) max {k M (0 ,x ) k L ( F ) , } , and the notion of unconditional convergence is weakened to convergence in the C´esarosense. Note that σ F = ω F if F is solid. Furthermore, an example (Proposition 5.17)shall show that unconditional convergence cannot longer be expected in the setting ofTMIB. We will also prove an analogue of Theorem 1.1 for DTMIB. Similarly as in thesolid case, but now by (1.2) instead of (1.1), the essential problem becomes to showthat the mappings F → F σd (Λ) , G ( G λ )) λ ∈ Λ and F σd (Λ) → F, ( c λ ) λ ∈ Λ X λ ∈ Λ c λ T σλ Ψare well-defined and continuous, and that the series P λ ∈ Λ c λ T λ Ψ is C´esaro summablein F for each c ∈ F σd (Λ), where Φ , Ψ belong to suitable function spaces on R n .As an application, we mention that our main results may be used to give explicitdescriptions of modulation spaces associated to TMIB and DTMIB. For example, (1.3)implies that M L b ⊗ π L = F M , (cf. Corollary 6.10). This identity and various relatedstatements were recently shown in [16] via different methods. We believe that our workmight be used to improve some of the results from [16] and we plan to investigate thisin the future (see also Problem 5.29).The paper is organised as follows. In the preliminary Sections 2 and 3, we fix thenotation and collect several results concerning TMIB and DTMIB. In Section 4, wedefine and discuss the twisted translation and the twisted convolution with respect toa real-valued n × n -matrix; although we are mainly interested in T σ and Notation
We use standard notation from distribution theory [25]. For a compact set K ⋐ R n we denote by D K the Fr´echet space of smooth functions ϕ on R n with supp ϕ ⊆ K .Given an open set U ⊆ R n , we define D ( U ) := lim −→ K ⋐ U D K . We write S ( R n ) for the Fr´echet space of rapidly decreasing smooth functions on R n and use the following family of norms on S ( R n ) k ϕ k S N := max | α |≤ N sup x ∈ R n | ∂ α ϕ ( x ) | (1 + | x | ) N , N ∈ N . The dual spaces D ′ ( R n ) and S ′ ( R n ) are the space of distributions on R n and the spaceof tempered distributions on R n , respectively. Unless stated otherwise, we endow thesespaces with their strong topology. ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 5
The constants in the Fourier transform are fixed as follows F ( f )( ξ ) = b f ( ξ ) := Z R n f ( x ) e − πix · ξ d x, f ∈ L ( R n ) . The Fourier transform is a topological isomorphism from S ( R n ) onto itself and extendsvia duality to a topological isomorphism from S ′ ( R n ) onto itself. Given a Banach space X ⊂ S ′ ( R n ), we define its associated Fourier space as the Banach space F X := { f ∈S ′ ( R n ) | F − f ∈ X } with norm k f k F X := kF − f k X .The translation and modulation operators are defined as T x f ( t ) = f ( t − x ) and M ξ f ( t ) = f ( t ) e πit · ξ , x, ξ ∈ R n . They act continuously on D ( R n ) and S ( R n ), and, byduality, therefore also on D ′ ( R n ) and S ′ ( R n ). We have that M ξ T x = e πix · ξ T x M ξ , F T x = M − x F , F M ξ = T ξ F . Furthermore, we write ˇ f ( t ) = f ( − t ) for reflection about the origin.Let Ω be a locally compact, σ -compact Hausdorff space and let (Ω , Σ , µ ) be a measurespace with µ a positive locally finite Borel measure. A Banach space E is called a solidBanach function space on Ω (cf. [13]) if E ⊂ L (Ω) with continuous inclusion and E satisfies the following condition: ∀ f ∈ E ∀ g ∈ L (Ω) : | g | ≤ | f | a.e. ⇒ g ∈ E and k g k E ≤ k f k E . Throughout the article,
C, C ′ , . . . denote absolute constants that may vary from placeto place.3. Translation-modulation invariant Banach spaces of distributionsand their duals
Definition and basic properties.
We start with the following basic definitionfrom [10].
Definition 3.1.
A Banach space E is called a translation-modulation invariant Banachspace of distributions (TMIB) on R n if the following three conditions hold:( i ) E satisfies the dense continuous inclusions S ( R n ) ֒ → E ֒ → S ′ ( R n ).( ii ) T x ( E ) ⊆ E and M ξ ( E ) ⊆ E for all x, ξ ∈ R n .( iii ) There exist τ j , C j > j = 0 ,
1, such that(3.1) ω E ( x ) := k T x k L ( E ) ≤ C (1 + | x | ) τ and ν E ( ξ ) := k M − ξ k L ( E ) ≤ C (1 + | ξ | ) τ ;for x, ξ ∈ R n fixed, the mappings T x : E → E and M ξ : E → E are continuousby the closed graph theorem.In what follows, the constants τ j , C j > j = 0 ,
1, will always refer to those occurringin (3.1).Let E be a TMIB. Then, E is separable and, for e ∈ E fixed, the mappings(3.2) R n → E, x T x e and R n → E, ξ M ξ e are continuous. The functions ω E and ν E are Borel measurable (as E is separable) andsubmultiplicative. A. DEBROUWERE AND B. PRANGOSKI
An interesting feature of TMIB is that they are stable under taking completed tensorproducts with respect to the π - and ǫ -topology [24]. Namely, let E j be a TMIB on R n j for j = 1 ,
2. Let τ denote either π or ǫ . Then, [10, Theorem 3.6] (and [16, Lemma 2.3]for τ = π ) yields that E b ⊗ τ E is a TMIB on R n + n with ω E b ⊗ τ E = ω E ⊗ ω E and ν E b ⊗ τ E = ν E ⊗ ν E .Next, we introduce dual translation-modulation invariant Banach spaces of distribu-tions [10]. Definition 3.2.
A Banach space is called a dual translation-modulation invariant Ba-nach space of distributions (DTMIB) on R n if it is the strong dual of a TMIB on R n .Let E be a DTMIB. Then, E satisfies the continuous inclusions S ( R n ) → E →S ′ ( R n ) and the conditions ( ii ) and ( iii ) from Definition 3.1. If E = E ′ , where E is a TMIB, then ω E = ˇ ω E and ν E = ν E , whence ω E and ν E are Borel measurable.Moreover, for e ∈ E fixed, the mappings in (3.2) are continuous with respect to theweak- ∗ topology on E . In general, E is not a TMIB. More precisely, the inclusion S ( R n ) → E need not be dense and the mappings in (3.2) may fail to be continuous;consider, e.g., E = L ∞ . However, if E is reflexive, then E is in fact a TMIB [9,Proposition 3.14] (see also [10, p. 827]).We now give some examples of TMIB and DTMIB; see also [10, Section 3]. Examples 3.3. ( i ) A Banach space E is called a solid TMIB (DTMIB) on R n if E isboth a TMIB (DTMIB) and a solid Banach function space on R n (with respect to theLebesgue measure). Then, k M ξ e k E = k e k E for all e ∈ E and ξ ∈ R n . A measurablefunction w : R n → (0 , ∞ ) is called a polynomially bounded weight function on R n ifthere are C, τ > w ( x + y ) ≤ Cw ( x )(1 + | y | ) τ , x, y ∈ R n . For 1 ≤ p ≤ ∞ we define L pw = L pw ( R n ) as the Banach space consisting of all (equiv-alence classes of) measurable functions f on R n such that k f k L pw := k f w k L p < ∞ .We define C ,w = C ,w ( R n ) as the closed subspace of L ∞ w consisting of all f ∈ C ( R n )such that lim | x |→∞ f ( x ) w ( x ) = 0. Then, L pw , 1 ≤ p < ∞ , is a solid TMIB, L pw ,1 < p ≤ ∞ , is a solid DTMIB, and C ,w is a TMIB. Similarly, we may considerweighted mixed-norm spaces. Let w be a polynomially bounded weight function on R n + n . For 1 ≤ p , p ≤ ∞ we define L p ,p w = L p ,p w ( R n + n ) as the Banach spaceconsisting of all (equivalence classes of) measurable functions f on R n + n such that k f k L p ,p w := k f w k L p ,p . Then, L p ,p w is a solid TMIB if 1 ≤ p , p < ∞ and a solidDTMIB if 1 < p , p ≤ ∞ .( ii ) Let E be a TMIB (DTMIB). Then, F E is a TMIB (DTMIB) with ω F E = ˇ ν E and ν F E = ω E . If E is solid, we have that k T x e k F E = k e k F E for all e ∈ F E and x ∈ R n .The Sobolev spaces F L pw , with w a polynomially bounded weight function on R n , areof this type.( iii ) Let w be a polynomially bounded weight function on R n and let E be a TMIB on R n . Then, the weighted Bochner-Lebesgue space L pw ( E ) = L pw ( R n ; E ), 1 ≤ p < ∞ ,and the weighted vector-valued C -space C ,w ( E ) = C ,w ( R n ; E ) are TMIB on R n + n . ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 7 If E ′ satisfies the Radon-Nikod´ym property (in particular, if E is reflexive), then (cf.[3, Theorem 3.5]) L pw ( E ′ ) = ( L q /w ( E )) ′ , < p ≤ ∞ , where q denotes the H¨older conjugate index to p . In particular, L pw ( E ′ ), 1 < p ≤ ∞ , isa DTMIB on R n + n .( iv ) The spaces L p ( R n ) b ⊗ π L p ( R n ) and L p ( R n ) b ⊗ ǫ L p ( R n ), 1 ≤ p , p < ∞ , areTMIB on R n + n consisting of locally integrable functions. In [10, Remark 3.10] it isshown that L p ( R n ) b ⊗ π L p ( R n ), 1 < p ≤
2, is not solid.3.2.
Convolution and multiplication.
Every TMIB or DTMIB E is a Banach con-volution module over the Beurling algebra L ω E and a Banach multiplication moduleover the Wiener-Beurling algebra F L ν E . More precisely, if E is a TMIB, the convolu-tion ∗ : S ( R n ) × S ( R n ) → S ( R n ) and multiplication · : S ( R n ) × S ( R n ) → S ( R n ) extenduniquely to continuous bilinear mappings ∗ : E × L ω E → E and · : E × F L ν E → E such that(3.3) k e ∗ f k E ≤ k e k E k f k L ωE , e ∈ E, f ∈ L ω E , and(3.4) k e · f k E ≤ k e k E k f k F L νE , e ∈ E, f ∈ F L ν E . Moreover, the following integral representations hold(3.5) e ∗ f = Z R n T x ef ( x )d x, e ∈ E, f ∈ L ω E , and(3.6) e · f = Z R n M − x e F − f ( x )d x, e ∈ E, f ∈ F L ν E , where the integrals should be interpreted as E -valued Bochner integrals [10, Proposition3.2]. Next, suppose that E is a DTMIB with E = E ′ , where E is a TMIB. Theconvolution and multiplication on E are defined via duality, namely, for e ∈ E , f ∈ L ω E ,and g ∈ F L ν E , we set h e ∗ f, g i := h e, g ∗ ˇ f i , g ∈ E , and h e · f, g i := h e, g · f i , g ∈ E . Then, the inequalities (3.3) and (3.4) hold true and the integral representations (3.5)and (3.6) are valid if the integrals are interpreted as E -valued Pettis integrals withrespect to the weak- ∗ topology on E [10, Corollary 3.5]. Hence, TMIB and DTMIBmay be viewed as Banach spaces of distributions having two module structures in thesense of [2].The goal of this subsection is to extend the previous results by showing that TMIBand DTMIB are in fact Banach convolution and multiplication modules over a certainweighted space of Radon measures and its associated Fourier space, respectively. Our The Wiener-Beurling algebra F L ν E is sometimes denoted as A ν E . A. DEBROUWERE AND B. PRANGOSKI approach is based on the integral representations (3.5) and (3.6). The following lemmawill allow us to treat TMIB and DTMIB simultaneously. Its proof is standard andtherefore we omit it.
Lemma 3.4.
Let X be a separable Banach space and set X = X ′ . Let (Ω , Σ , µ ) bea measure space with µ a complex measure. Let f : Ω → X be weak- ∗ measurable,i.e., the function Ω → C , x
7→ h f ( x ) , g i is measurable for every g ∈ X . Furthermore,suppose that (3.7) Z Ω k f ( x ) k X d | µ | ( x ) < ∞ . Then, f : Ω → X is Pettis integrable with respect to the weak- ∗ topology on X and (3.8) (cid:13)(cid:13)(cid:13)(cid:13)Z Ω f ( x )d µ ( x ) (cid:13)(cid:13)(cid:13)(cid:13) X ≤ Z Ω k f ( x ) k X d | µ | ( x ) . We will use Lemma 3.4 without explicitly referring to it.Let ω : R n → [1 , ∞ ) be a Borel measurable submultiplicative polynomially boundedfunction. We denote by M ω = M ω ( R n ) the Banach space consisting of all complexRadon measures µ on R n such that k µ k M ω := R R n ω ( x )d | µ | ( x ) < ∞ . The space M ω ⊂S ′ ( R n ) is a Banach convolution module and its associated Fourier space F M ω is aBanach multiplication module if the multiplication is defined via the Fourier transformand the convolution in M ω . Since M ω ⊆ M , the elements of F M ω are boundedcontinuous functions and the multiplication defined above coincides with the ordinarymultiplication of continuous functions.Let E be a TMIB or a DTMIB and set e ω E = max { , ω E } . We define the convolutionof e ∈ E and µ ∈ M e ω E as e ∗ µ := Z R n T x e d µ ( x ) ∈ E, where the integral should be interpreted as an E -valued Bochner integral if E is aTMIB and as an E -valued Pettis integral with respect to the weak- ∗ topology on E if E is a DTMIB; hereafter, for DTMIB E , E -valued Pettis integrals will always be meantwith respect to the weak- ∗ topology on E (cf. Lemma 3.4). Hence, ∗ : E × M e ω E → E is a continuous bilinear mapping such that k e ∗ µ k E ≤ k e k E k µ k M e ωE , e ∈ E, µ ∈ M e ω E . If d µ ( x ) = f ( x )d x with f ∈ L e ω E , this definition of convolution coincides with the onegiven at the beginning of the subsection. Furthermore, if Z R n (1 + | x | ) N d | µ | ( x ) < ∞ , ∀ N ∈ N , then µ ∈ O ′ C ( R n ) [25, p. 244] and h e ∗ µ, ϕ i = h e, ϕ ∗ ˇ µ i , ϕ ∈ S ( R n ) , ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 9 whence e ∗ µ is equal to the S ′ ( R n ) × O ′ C ( R n )-convolution of e and µ [25, Theor`emeXI, p. 247]. Next, we consider multiplication. Set e ν E = max { , ν E } . We define themultiplication of e ∈ E and f ∈ F M e ν E as e · f := Z R n M − x e d F − f ( x );the integral should be interpreted as an E -valued Bochner integral if E is a TMIBand as an E -valued Pettis integral if E is a DTMIB. Hence, · : E × F M e ν E → E is acontinuous bilinear mapping such that k e · f k E ≤ k e k E k f k FM e νE , e ∈ E, f ∈ F M e ν E . If f ∈ F L e ν E , this definition of multiplication coincides with the one given at thebeginning of the subsection. Furthermore, if Z R n (1 + | x | ) N d |F − f | ( x ) < ∞ , ∀ N ∈ N , then f ∈ O M ( R n ) [25, p. 243] and h e · f, ϕ i = h e, ϕ · f i , ϕ ∈ S ( R n ) , whence e · f is equal to the S ′ ( R n ) × O M ( R n )-multiplication of e and f [25, Theor`emeX, p. 246]. Suppose that E is a DTMIB with E = E ′ , where E is a TMIB. For e ∈ E , µ ∈ M e ω E , and f ∈ F M e ν E it holds that h e ∗ µ, g i = h e, g ∗ ˇ µ i and h e · f, g i = h e, g · f i , g ∈ E . Every solid Banach function space is a Banach multiplication module over L ∞ . Wenow use the previous observations to formulate a result that, for our purposes, willturn out to be the suitable analogue of this fact for TMIB and DTIMB. We first needto introduce some terminology. A lattice Λ is a discrete subgroup of R n that spans thereal vector space R n . There is a unique invertible n × n -matrix A Λ such that Λ = A Λ Z n .The dual lattice of Λ is defined as Λ ⊥ = ( A t Λ ) − Z n = { µ ∈ R n | λ · µ ∈ Z , ∀ λ ∈ Λ } . Wedefine I Λ := A Λ [0 , n and vol(Λ) := | I Λ | = | det A Λ | . Lemma 3.5.
Let ω : R n → [1 , ∞ ) be a Borel measurable submultiplicative polynomiallybounded function. Let Λ be a lattice in R n . Then, for every y ∈ R n , the bilinearmapping F L ω × S ( R n ) → F M ω , ( f, ϕ ) X λ ∈ Λ e πiy · λ T λ ( f ϕ ) , is well-defined and continuous. Furthermore, there are C > and N ∈ N such that sup y ∈ R n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X λ ∈ Λ e πiy · λ T λ ( f ϕ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) FM ω ≤ C k f k F L ω k ϕ k S N , f ∈ F L ω , ϕ ∈ S ( R n ) . Proof.
Let f ∈ F L ω , ϕ ∈ S ( R n ), and y ∈ R n be arbitrary. The Poisson summationformula implies that(3.9) F − X λ ∈ Λ e πiy · λ T λ ( f ϕ ) ! = 1vol(Λ) X µ ∈ Λ ⊥ c f ϕ ( µ + y ) T − µ − y δ in S ′ ( R n ) . Hence, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X λ ∈ Λ e πiy · λ T λ ( f ϕ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) FM ω = 1vol(Λ) X µ ∈ Λ ⊥ | b f ∗ b ϕ ( µ + y ) | ˇ ω ( µ + y ) ≤ Z R n | b f ( y − x ) | ˇ ω ( y − x ) X µ ∈ Λ ⊥ | b ϕ ( µ + x ) | ˇ ω ( µ + x )d x ≤ C k b ϕ k L ∞ (1+ |·| ) n +1 ˇ ω kF f k L ω . As the Fourier transform is an isomorphism from S ( R n ) onto itself and kF f k L ω = k f k F L ω , this completes the proof. (cid:3) Corollary 3.6.
Let Λ be a lattice in R n and let ϕ ∈ S ( R n ) . Then, for every y ∈ R n ,the bilinear mapping E × F L e ν E → E, ( e, f ) e · X λ ∈ Λ e πiy · λ T λ ( f ϕ ) , is well-defined and continuous. Furthermore, there is C > such that sup y ∈ R n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e · X λ ∈ Λ e πiy · λ T λ ( f ϕ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ≤ C k e k E k f k F L e νE , e ∈ E, f ∈ F L e ν E . Amalgam spaces.
In this subsection, we define amalgam spaces which have aTMIB or a DTMIB as local component. These spaces will play an important technicalrole in the rest of this article. We refer to [11, 16] for more information.Let E be a TMIB or DTMIB. We define E loc = { f ∈ D ′ ( R n ) | χf ∈ E, ∀ χ ∈ D ( R n ) } .Since D ( R n ) ⊂ F L ν E , the function R n → E , x f T x χ is continuous for all f ∈ E loc and χ ∈ D ( R n ). Let w be a polynomially bounded weight function on R n and let1 ≤ p ≤ ∞ . Fix χ ∈ D ( R n ) \{ } . We define the amalgam space W ( E, L pw ) as the spaceconsisting of all f ∈ E loc such that (cf. [11], [16, Section 3]) k f k W ( E,L pw ) := (cid:18)Z R n k f T x χ k pE w ( x ) p d x (cid:19) /p < ∞ (with the obvious modification for p = ∞ ). Then, W ( E, L pw ) is a Banach space whosedefinition is independent of the choice χ ∈ D ( R n ) \{ } and different non-zero elementsof D ( R n ) induce equivalent norms on W ( E, L pw ) (cf. [16, Lemma 3.4], [11, Theorem 1]).By [16, Lemma 3.2], W ( E, L pw ), 1 ≤ p < ∞ , is a TMIB if E is so, while W ( E, L pw ),1 ≤ p ≤ ∞ , is a DTMIB if E is so. ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 11 The twisted translation an the twisted convolution
Fix a real-valued n × n -matrix B . For x ∈ R n we define the twisted translation withrespect to B as T Bx f ( t ) := T x M − Bx f ( t ) = f ( t − x ) e − πiBx · ( t − x ) , f ∈ D ′ ( R n ) . Note that T x = T x . For all x, y ∈ R n , f ∈ D ′ ( R n ), and ϕ ∈ C ∞ ( R n ) it holds that( i ) T Bx T B t y = T B t y T Bx .( ii ) T Bx ( f · ϕ ) = T Bx f · T x ϕ = T x f · T Bx ϕ .( iii ) T Bx f · T − Bx ϕ = T x ( f · ϕ ).We define the twisted convolution with respect to B of f, g ∈ L as f ∗ B g ( t ) := Z R n f ( x ) T Bx g ( t )d x. Note that f ∗ g = f ∗ g . Define θ B ( f )( x ) := e πiBx · x f ( x ) , f ∈ D ′ ( R n ) . For all f, g ∈ L , h ∈ L ∩ L ∞ , it holds that( i ) f ∗ B g = g ∗ B t f .( ii ) f ∗ B g ( t ) = Z R n f ( x ) T − Bt ( θ B (ˇ g ))( x )d x .( iii ) Z R n f ∗ B g ( t ) h ( t )d t = Z R n f ( t ) h ∗ − B θ B (ˇ g )( t )d t . Definition 4.1.
Consider the real-valued 2 n × n -matrix B := (cid:18) I (cid:19) . Following the notation used in the introduction, we set T σ ( x,ξ ) f ( t, η ) := T B ( x,ξ ) f ( t, η ) = f ( t − x, η − ξ ) e − πix · ( η − ξ ) , ( x, ξ ) ∈ R n , and for f, g ∈ L ( R n ) f g ( t, η ) := f ∗ B g ( t, η ) = Z Z R n f ( x, ξ ) g ( t − x, η − ξ ) e − πix · ( η − ξ ) d x d ξ. Next, we extend the twisted convolution to S ′ ( R n ). The proof of the following lemmais straightforward and we omit it. Lemma 4.2. ( i ) The mapping T Bx : S ( R n ) → S ( R n ) is continuous for each x ∈ R n . Moreprecisely, k T Bx ϕ k S N ≤ (1 + 2 π k B k ) N k ϕ k S N (1 + | x | ) N , ϕ ∈ S ( R n ) , N ∈ N , where k B k denotes the operator norm of B . ( ii ) The mapping R n → S ( R n ) , x T Bx ϕ is continuous for each ϕ ∈ S ( R n ) . ( iii ) The mappings θ B : S ( R n ) → S ( R n ) and θ B : S ′ ( R n ) → S ′ ( R n ) are continuous. ( iv ) The bilinear mapping ∗ B : S ( R n ) × S ( R n ) → S ( R n ) is continuous. We define the twisted convolution of f ∈ S ′ ( R n ) and ϕ ∈ S ( R n ) as(4.1) f ∗ B ϕ ( x ) := h f, T − Bx θ B ( ˇ ϕ ) i . Then, f ∗ B ϕ ∈ C ( R n ) and k f ∗ B ϕ k L ∞ (1+ | · | ) − N < ∞ for some N ∈ N . If A ⊂ S ′ ( R n ) isbounded, the previous estimate holds uniformly for f ∈ A . Since S ′ ( R n ) is bornological,this implies that the mapping S ′ ( R n ) → S ′ ( R n ) , f f ∗ B ϕ is continuous. As L is dense in S ′ ( R n ), we have that h f ∗ B ϕ, ψ i = h f, ψ ∗ − B θ B ( ˇ ϕ ) i , ψ ∈ S ( R n ) , for all f ∈ S ′ ( R n ) and ϕ ∈ S ( R n ).Finally, we discuss the twisted convolution on TMIB and DTMIB. Let E be a TMIBor a DTMIB. Then, T Bx : E → E is continuous for each x ∈ R n and(4.2) ρ BE ( x ) := k T Bx k L ( E ) ≤ ω E ( x ) ν E ( Bx ) ≤ C (1 + | x | ) τ + τ , where C = C C max { , k B k τ } . Note that ρ BE is submultiplicative and polynomiallybounded. For e ∈ E fixed, the mapping(4.3) R n → E, x T Bx e, is continuous if E is a TMIB and continuous with respect to the weak- ∗ topology on E if E is a DTMIB. Consequently, ρ BE is Borel measurable when E is a TMIB (as E is separable). If E is a DTMIB with E = E ′ , where E a TMIB, the bipolar theoremyields that ρ BE = ˇ ρ − BE , whence ρ BE is Borel measurable in this case as well.Given a Banach space X ⊂ S ′ ( R n ), we define the Banach spaces ˇ X := { f ∈S ′ ( R n ) | ˇ f ∈ X } with norm k f k ˇ X := k ˇ f k X and θ B X = { f ∈ S ′ ( R n ) | θ − B f ∈ X } with norm k f k θ B X := k θ − B f k X . Furthermore, given a polynomially bounded weightfunction w on R n , we denote by C w = C w ( R n ) the space L ∞ w ∩ C ( R n ); of course, it isa closed subspace of L ∞ w .Assume that E is a TMIB. The twisted convolution of e ∈ E and g ∈ ( θ − B E ′ )ˇ isdefined as e ∗ B g ( x ) := E h e, T − Bx θ B (ˇ g ) i E ′ . Similarly, we define the twisted convolution of e ∈ E ′ and g ∈ ( θ − B E )ˇas e ∗ B g ( x ) := E ′ h e, T − Bx θ B (ˇ g ) i E . Obviously, these definitions coincide with the one given in (4.1) if g ∈ S ( R n ). Notethat the bilinear mappings ∗ B : E × ( θ − B E ′ )ˇ → C / ˇ ρ BE and ∗ B : E ′ × ( θ − B E )ˇ → C / ˇ ρ BE ′ are well-defined and continuous.5. Discrete spaces associated to TMIB and DTMIB
Throughout this section, E always stands for a TMIB or a DTMIB. We also fix areal-valued n × n -matrix B , a lattice Λ in R n , and a bounded open neighbourhood U of the origin such that the family of sets { λ + U | λ ∈ Λ } is pairwise disjoint. ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 13
Definition and basic properties.
The following fundamental definition is in-spired by [13, Definition 3.4], where a discrete space is associated to a solid Banachfunction space.
Definition 5.1.
Let χ ∈ D ( U ) \{ } . We define the discrete space associated to E withrespect to B as E Bd (Λ) = E Bd,χ (Λ) := ( c = ( c λ ) λ ∈ Λ ∈ C Λ (cid:12)(cid:12)(cid:12) S χ ( c ) := X λ ∈ Λ c λ T Bλ χ ∈ E ) and endow it with the norm k c k E Bd (Λ) = k c k E Bd,χ (Λ) := k S χ ( c ) k E .We start by showing that E Bd (Λ) is a Banach space whose definition is independentof χ ∈ D ( U ) \{ } . Theorem 5.2. ( i ) E Bd (Λ) is a Banach space. ( ii ) The definition of E Bd (Λ) is independent of the choice χ ∈ D ( U ) \{ } and differ-ent non-zero elements of D ( U ) induce equivalent norms on E Bd (Λ) .Proof. ( i ) Let ( c j ) j ∈ N be a Cauchy sequence in E Bd (Λ). Since E is continuously includedin D ′ ( R n ), the inclusion mapping E Bd (Λ) → C Λ is continuous. Hence, there is c ∈ C Λ such that lim j →∞ c j = c in C Λ . As ( S χ ( c j )) j ∈ N is a Cauchy sequence in E , there is e ∈ E such that lim j →∞ S χ ( c j ) = e in E . Note that e = S χ ( c ) in D ′ ( R n ). Therefore, c ∈ E Bd (Λ) and lim j →∞ c j = c in E Bd (Λ).( ii ) We divide the proof into three steps.STEP I: Let e χ ∈ D ( U ) \{ } be such that e χ = 1 on some non-empty open subset V of U . Then, E Bd, e χ (Λ) is continuously included into E Bd,χ (Λ) for all χ ∈ D ( U ) \{ } . Let x ∈ U and r > x − r, x + r ] n ⊂ V . Pick ψ ∈ D [ − r,r ] n suchthat P m ∈ Z n T rm ψ = 1 on R n . Hence, there is N ∈ N such that P | m |≤ N T rm ψ = 1 onsupp χ . For all c ∈ E Bd, e χ (Λ) it holds that X λ ∈ Λ c λ T Bλ χ = X | m |≤ N X λ ∈ Λ c λ T Bλ ( χT rm ψ )= X | m |≤ N X λ ∈ Λ c λ e πiBλ · ( x − rm ) T rm − x T Bλ ( T x ψT x − rm χ )= X | m |≤ N T rm − x X λ ∈ Λ c λ e πiB t ( x − rm ) · λ T Bλ ( e χT x ψT x − rm χ ) ! = X | m |≤ N T rm − x X λ ∈ Λ c λ T Bλ e χ · X λ ′ ∈ Λ e πiB t ( x − rm ) · λ ′ T λ ′ ( T x − rm χT x ψ ) ! . The result is therefore a consequence of Corollary 3.6.STEP II:
Let χ ∈ D ( U ) \{ } . Choose e χ ∈ D ( U ) such that supp e χ ⊂ { x ∈ U | χ ( x ) =0 } and e χ = 1 on some non-empty open subset V of U . Then, E Bd,χ (Λ) = E Bd, e χ (Λ) withequivalent norms. By STEP I, E Bd, e χ (Λ) is continuously included in E Bd,χ (Λ). We now show the converseinclusion. Set ϕ = e χ/χ ∈ D ( U ). For all c ∈ E Bd,χ (Λ) it holds that X λ ∈ Λ c λ T Bλ e χ = X λ ∈ Λ c λ T Bλ ( χϕ ) = X λ ∈ Λ c λ T Bλ χ · X λ ′ ∈ Λ T λ ′ ϕ, whence the result follows from Corollary 3.6.STEP III: Let χ , χ ∈ D ( U ) \{ } . Then, E Bd,χ (Λ) = E Bd,χ (Λ) with equivalentnorms. Choose e χ , e χ ∈ D ( U ) as in STEP II. Then, E Bd,χ (Λ) = E Bd, e χ (Λ) = E Bd, e χ (Λ) = E Bd,χ (Λ)with equivalent norms, where the first and third equality follow from STEP II and thesecond equality follows from STEP I. (cid:3) Remark . An immediate consequence of Theorem 5.2 is that E Bd (Λ) also does notdepend on the bounded open set U as long as the family of sets { λ + U | λ ∈ Λ } ispairwise disjoint.The next result, which will be used later on, follows from an inspection of the proofof Theorem 5.2. Lemma 5.4.
Let A ⊂ D ( U ) \{ } be a bounded subset of D ( U ) . ( i ) For every χ ∈ D ( U ) \{ } there is C > such that sup ϕ ∈ A k c k E Bd,ϕ (Λ) ≤ C k c k E Bd,χ (Λ) , c ∈ E Bd (Λ) . ( ii ) Suppose that there is a non-empty open subset V of U such that inf ϕ ∈ A inf x ∈ V | ϕ ( x ) | > . Then, for every χ ∈ D ( U ) \{ } there is C > such that k c k E Bd,χ (Λ) ≤ C inf ϕ ∈ A k c k E Bd,ϕ (Λ) , c ∈ E Bd (Λ) . Consider the following discrete spaces S d (Λ) := { c ∈ C Λ | k c k S Nd (Λ) := sup λ ∈ Λ | c λ | (1 + | λ | ) N < ∞ , ∀ N ∈ N } and S ′ d (Λ) := { c ∈ C Λ | ∃ N ∈ N : k c k S − Nd (Λ) := sup λ ∈ Λ | c λ | (1 + | λ | ) − N < ∞} , and endow them with their natural Fr´echet space and ( LB )-space topology, respec-tively. The strong dual of S d (Λ) may be topologically identified with S ′ d (Λ). We thenhave: Proposition 5.5.
The following continuous inclusions hold S d (Λ) → E Bd (Λ) → S ′ d (Λ) . In view of the continuous inclusions S ( R n ) → E → S ′ ( R n ), Proposition 5.5 is adirect consequence of the next lemma. ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 15
Lemma 5.6.
Let ϕ ∈ S ( R n ) . ( i ) The mapping S d (Λ) → S ( R n ) , c X λ ∈ Λ c λ T Bλ ϕ is well-defined and continuous, and the series P λ ∈ Λ c λ T Bλ ϕ is absolutely sum-mable in S ( R n ) . ( ii ) The mapping (5.1) S ′ d (Λ) → S ′ ( R n ) , c X λ ∈ Λ c λ T Bλ ϕ is well-defined and continuous, and the series P λ ∈ Λ c λ T Bλ ϕ is absolutely sum-mable in S ′ ( R n ) . ( iii ) Suppose that ϕ ∈ D ( U ) \{ } . Then, c ∈ C Λ belongs to S ′ d (Λ) if and onlyif P λ ∈ Λ c λ T Bλ ϕ ∈ S ′ ( R n ) . Moreover, the mapping in (5.1) is a topologicalembedding.Proof. Parts ( i ) and ( ii ) are easy consequences of Lemma 4.2 and we omit their proofs.We now show ( iii ). Let c ∈ C Λ be such that P λ ∈ Λ c λ T Bλ ϕ ∈ S ′ ( R n ). Hence, there are N ∈ N and C > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)*X λ ∈ Λ c λ T Bλ ϕ, ψ +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ψ k S N , ψ ∈ D ( R n ) . Pick ψ ∈ D ( U ) such that R R n ϕ ( x ) ψ ( x )d x = 1. Then, *X λ ′ ∈ Λ c λ ′ T Bλ ′ ϕ, T − Bλ ψ + = c λ Z R n T Bλ ϕ ( x ) T − Bλ ψ ( x )d x = c λ , λ ∈ Λ . Lemma 4.2( i ) now implies that, for all λ ∈ Λ, | c λ | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)*X λ ′ ∈ Λ c λ ′ T Bλ ′ ϕ, T − Bλ ψ +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k T − Bλ ψ k S N ≤ C (1 + 2 π k B k ) N k ψ k S N (1 + | λ | ) N , whence c ∈ S ′ d (Λ). Finally, we show that the continuous mapping (5.1) is a topologicalembedding. It is clear that this mapping is injective and, by what we have just shown,it also has closed range. Since S ′ ( R n ) is a ( DF S )-space and a closed subspace of a(
DF S )-space is again a (
DF S )-space, we obtain that the range of the mapping (5.1)is a (
DF S )-space. Hence, the result follows from the De Wilde open mapping theorem[23, Theorem 1, p. 59] (cf. [23, Theorem 8, p. 63]). (cid:3)
Next, we give two results that will play a crucial role in the rest of the article. Thefollowing result is the analogue of [13, Proposition 5.2] in our setting (see also the proofof [19, Theorem 12.2.4]).
Proposition 5.7.
The bilinear mapping E Bd (Λ) × S ( R n ) → E, ( c, ϕ ) S ϕ ( c ) , with S ϕ ( c ) = X λ ∈ Λ c λ T Bλ ϕ, is well-defined and continuous and uniquely extends to a continuous bilinear mapping (5.2) E Bd (Λ) × W ( F L e ν E , L ω E ) → E, ( c, ϕ ) e S ϕ ( c ) . Furthermore, if E is a DTMIB with E = E ′ , where E is a TMIB, there is χ ∈D ( U ) \{ } such that for every g ∈ E and ϕ ∈ W ( F L e ν E , L ω E ) there is h ∈ E such that (5.3) h e S ϕ ( c ) , g i = h S χ ( c ) , h i , c ∈ E Bd (Λ) . Proof.
Let r > − r, r ] n ⊂ U and let χ ∈ D [ − r,r ] n be such that P m ∈ Z n T rm χ = 1 on R n . Choose ψ ∈ D [ − r, r ] n such that ψ = 1 on [ − r, r ] n and ψ ∈ D [ − r, r ] n such that ψ = 1 on [ − r, r ] n . Let c ∈ E Bd (Λ) and ϕ ∈ W ( F L e ν E , L ω E )be arbitrary. For each m ∈ Z n , we infer X λ ∈ Λ c λ T Bλ ( ϕT rm χ ) = X λ ∈ Λ c λ e − πiBλ · rm T rm T Bλ (( T − rm ϕ ) χ )(5.4) = T rm X λ ∈ Λ c λ e − πiB t rm · λ T Bλ ( χ ( T − rm ϕ ) ψ )= T rm X λ ∈ Λ c λ T Bλ χ · X λ ′ ∈ Λ e − πiB t rm · λ ′ T λ ′ (( ψT − rm ϕ ) ψ ) ! . Hence, Corollary 3.6 yields that P λ ∈ Λ c λ T Bλ ( ϕT rm χ ) ∈ E and that X m ∈ Z n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X λ ∈ Λ c λ T Bλ ( ϕT rm χ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ≤ C k c k E Bd (Λ) X m ∈ Z n ω E ( rm ) k ψT − rm ϕ k F L e νE = C k c k E Bd (Λ) X m ∈ Z n ω E ( rm ) k ϕT rm ψ k F L e νE . Choose ψ ∈ D [ − r, r ] n such that ψ = 1 on [ − r, r ] n . Then, X m ∈ Z n ω E ( rm ) k ϕT rm ψ k F L e νE = r − n X m ∈ Z n Z rm +[ − r/ ,r/ n k ϕT rm ψ k F L e νE ω E ( rm )d x ≤ C ′ r − n X m ∈ Z n Z rm +[ − r/ ,r/ n k ϕT x ψ T rm ψ k F L e νE ω E ( x )d x ≤ C ′ r − n k ψ k F L e νE X m ∈ Z n Z rm +[ − r/ ,r/ n k ϕT x ψ k F L e νE ω E ( x )d x = C ′ r − n k ψ k F L e νE k ϕ k W ( F L e νE ,L ωE ) . We deduce that(5.5) X m ∈ Z n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X λ ∈ Λ c λ T Bλ ( ϕT rm χ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ≤ C ′′ k c k E Bd (Λ) k ϕ k W ( F L e νE ,L ωE ) . Now suppose that ϕ ∈ S ( R n ). Since the double series X m ∈ Z n , λ ∈ Λ c λ T Bλ ( ϕT rm χ ) ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 17 is absolutely summable in S ′ ( R n ), we have that (cf. Lemma 5.6( ii ))(5.6) X λ ∈ Λ c λ T Bλ ϕ = X m ∈ Z n X λ ∈ Λ c λ T Bλ ( ϕT rm χ ) in S ′ ( R n ) , c ∈ E Bd (Λ) , ϕ ∈ S ( R n ) . As S ( R n ) is dense in W ( F L e ν E , L ω E ), the first statement is therefore a consequence of(5.5). Moreover, we obtain that(5.7) e S ϕ ( c ) = X m ∈ Z n X λ ∈ Λ c λ T Bλ ( ϕT rm χ ) , c ∈ E Bd (Λ) , ϕ ∈ W ( F L e ν E , L ω E ) . Next, suppose that E is a DTMIB with E = E ′ , where E is a TMIB. Let g ∈ E and ϕ ∈ W ( F L e ν E , L ω E ) be arbitrary. Similarly as in the proof of (5.5), one can show thatthe series X m ∈ Z n T − rm g · X λ ′ ∈ Λ e − πiB t rm · λ ′ T λ ′ ( ψT − rm ϕ ) ! is absolutely summable in E ; denote it by h ∈ E . Then, (5.4) and (5.7) give (5.3). (cid:3) Corollary 5.8.
The space W ( F L e ν E , L ω E ) is continuously included into E . Conse-quently, W ( F L e ν E , L ω E ) ⊂ E ′ continuously if E is a TMIB and W ( F L e ν E , L ω E ) ⊂ E continuously if E is a DTMIB and E = E ′ , where E is a TMIB.Proof. Let c ∈ C Λ be such that c = 1 and c λ = 0 for λ ∈ Λ \{ } . Since S ( R n ) is densein W ( F L e ν E , L ω E ), Proposition 5.7 yields that e S ϕ ( c ) = ϕ for all ϕ ∈ W ( F L e ν E , L ω E ).The result now follows from another application of Proposition 5.7. (cid:3) Remark . From now on, we will denote the continuous extension e S ϕ ( c ) simply by S ϕ ( c ). We emphasise that, at the moment, we do not claim that S ϕ ( c ) is given by P λ ∈ Λ c λ T Bλ ϕ for general ϕ ∈ W ( F L e ν E , L ω E ) as we do not give any meaning to thisseries for such ϕ . Later on, we will prove that the series P λ ∈ Λ c λ T Bλ ϕ converges to S ϕ ( c ) in the C´esaro sense (see Corollary 5.23 below).We now show a sampling inequality for the twisted translation; it should be comparedwith [19, Lemma 3.9( a ) and Proposition 5.2] and [19, Proposition 11.1.4].Corollary 5.8 implies that the bilinear mapping(5.8) ∗ B : E × ( θ − B W ( F L e ν E , L ω E ))ˇ → C / ˇ ρ BE is well-defined and continuous (cf. the last part of Section 4). Proposition 5.10.
The bilinear mapping E × ( θ − B W ( F L e ν E , L ω E ))ˇ → E Bd (Λ) , ( e, ϕ ) R ϕ ( e ) := ( e ∗ B ϕ ( λ )) λ ∈ Λ , is well-defined and continuous. The proof of Proposition 5.10 is based on the identity shown in the next lemma.
Lemma 5.11.
For all f ∈ S ′ ( R n ) , ϕ ∈ S ( R n ) and χ ∈ D ( U ) , it holds that (5.9) X λ ∈ Λ f ∗ B ϕ ( λ ) T Bλ χ = Z R n T x f · X λ ∈ Λ e − πiB t x · λ T λ ( T x ( θ B ( ˇ ϕ )) χ )d x in S ′ ( R n ) , where the integral should be interpreted as an S ′ ( R n ) -valued Pettis integral with respectto the weak- ∗ topology on S ′ ( R n ) .Proof. Note that the mapping R n → D L ∞ (1+ |·| ) − ( R n ) , x X λ ∈ Λ e − πiB t x · λ T λ ( T x ( θ B ( ˇ ϕ )) χ )is continuous. This implies that the mapping R n → S ′ ( R n ) , x T x f · X λ ∈ Λ e − πiB t x · λ T λ ( T x ( θ B ( ˇ ϕ )) χ )is continuous with respect to the weak- ∗ topology on S ′ ( R n ). Hence, by Lemma 5.6( ii ),we only need to show that X λ ∈ Λ Z R n f ∗ B ϕ ( λ ) T Bλ χ ( x ) ψ ( x )d x = Z R n * T x f · X λ ∈ Λ e − πiB t x · λ T λ ( T x ( θ B ( ˇ ϕ )) χ ) , ψ + d x for all ψ ∈ S ( R n ). We have that X λ ∈ Λ Z R n f ∗ B ϕ ( λ ) T Bλ χ ( x ) ψ ( x )d x = X λ ∈ Λ h f ( t ) , T − Bλ ( θ B ( ˇ ϕ ))( t ) i Z R n T Bλ χ ( x ) ψ ( x )d x = X λ ∈ Λ (cid:28) f ( t ) , Z R n T − Bλ ( θ B ( ˇ ϕ ))( t ) T Bλ χ ( t + x ) ψ ( t + x )d x (cid:29) . As the function R n → C , ( t, x ) T − Bλ ( θ B ( ˇ ϕ ))( t ) T Bλ χ ( t + x ) ψ ( t + x ) , belongs to S ( R n ), we infer that X λ ∈ Λ Z R n f ∗ B ϕ ( λ ) T Bλ χ ( x ) ψ ( x )d x = X λ ∈ Λ h f ( t ) ⊗ x ) , T − Bλ ( θ B ( ˇ ϕ ))( t ) T Bλ χ ( t + x ) ψ ( t + x ) i = X λ ∈ Λ Z R n h f ( t ) , T − Bλ ( θ B ( ˇ ϕ ))( t ) T Bλ χ ( t + x ) ψ ( t + x ) i d x = Z R n X λ ∈ Λ h f, T − Bλ ( θ B ( ˇ ϕ )) T − x ( T Bλ χψ ) i d x = Z R n X λ ∈ Λ h T x f, T x T − Bλ ( θ B ( ˇ ϕ )) T Bλ χψ i d x = Z R n X λ ∈ Λ h T x f, e − πiB t x · λ T λ ( T x ( θ B ( ˇ ϕ )) χ ) ψ i d x ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 19 = Z R n * T x f · X λ ∈ Λ e − πiB t x · λ T λ ( T x ( θ B ( ˇ ϕ )) χ ) , ψ + d x. This completes the proof of the lemma. (cid:3)
Proof of Proposition 5.10. As e ∗ B ϕ is continuous, we can evaluate it at λ ∈ Λ. Fix χ ∈ D ( U ) \{ } . Since S ( R n ) is dense in W ( F L e ν E , L ω E ), Lemma 5.11 and the continuityof the mapping (5.8) imply that it suffices to show that the bilinear mapping E × ( θ − B W ( F L e ν E , L ω E ))ˇ → E, ( e, ϕ ) Z R n T x e · X λ ∈ Λ e − πiB t x · λ T λ ( T x ( θ B ( ˇ ϕ )) χ )d x (5.10)is well-defined and continuous, where the integral should be interpreted as an E -valuedBochner integral if E is a TMIB and as an E -valued Pettis integral if E is a DTMIB.Let e ∈ E and ϕ ∈ ( θ − B W ( F L e ν E , L ω E ))ˇ be arbitrary. Choose χ ∈ D ( U ) such that χ = 1 on supp χ . Then, X λ ∈ Λ e − πiB t x · λ T λ ( T x ( θ B ( ˇ ϕ )) χ ) = X λ ∈ Λ e − πiB t x · λ T λ ( T x ( θ B ( ˇ ϕ )) χχ ) , x ∈ R n . Hence, Corollary 3.6 verifies that, for x ∈ R n fixed, the integrand in (5.10) is a well-defined element of E . Claim.
The mapping(5.11) R n → E, x T x e · X λ ∈ Λ e − πiB t x · λ T λ ( T x ( θ B ( ˇ ϕ )) χ ) , is strongly measurable if E is a TMIB and weak- ∗ measurable if E is a DTMIB.Assuming the validity of the claim, Corollary 3.6 gives the bound Z R n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) T x e · X λ ∈ Λ e − πiB t x · λ T λ (( T x ( θ B ( ˇ ϕ )) χ ) χ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E d x ≤ C k e k E Z R n ω E ( x ) k T x ( θ B ( ˇ ϕ )) χ k F L e νE d x = C k e k E k ϕ k ( θ − B W ( F L e νE ,L ωE ))ˇ , whence the mapping (5.10) is well-defined and continuous. It remains to prove theclaim. First suppose that ϕ ∈ S ( R n ). For each x ∈ R n , it holds that X λ ∈ Λ e − πiB t x · λ T λ ( T x ( θ B ( ˇ ϕ )) χ ) ∈ D L ∞ ( R n ) and X λ ∈ Λ T λ ( T − B t x ( θ B ( ˇ ϕ )) χ ) ∈ D L ∞ ( R n ) . We infer that, for all ψ ∈ S ( R n ) and x ∈ R n , * T x e · X λ ∈ Λ e − πiB t x · λ T λ ( T x ( θ B ( ˇ ϕ )) χ ) , ψ + = * T x e, ψ X λ ∈ Λ e − πiB t x · λ T λ ( T x ( θ B ( ˇ ϕ )) χ ) + = * T x e, e πiB t x · x M − B t x ψ X λ ∈ Λ T λ ( T − B t x ( θ B ( ˇ ϕ )) χ ) !+ = * T B t x e · X λ ∈ Λ T λ ( T − B t x ( θ B ( ˇ ϕ )) χ ) , ψ + , and, consequently,(5.12) T x e · X λ ∈ Λ e − πiB t x · λ T λ ( T x ( θ B ( ˇ ϕ )) χ ) = T B t x e · X λ ∈ Λ T λ ( T − B t x ( θ B ( ˇ ϕ )) χ ) . Lemma 3.5 implies that X λ ∈ Λ T λ ( T − B t x ( θ B ( ˇ ϕ )) χ ) = X λ ∈ Λ T λ ( T − B t x ( θ B ( ˇ ϕ )) χχ ) ∈ F M e ν E and therefore the multiplication on the right-hand side in (5.12) may be interpreted asthe multiplication on E × F M e ν E . Since the mapping R n → F L e ν E , x T − B t x ( θ B ( ˇ ϕ )),is continuous, Lemma 3.5 gives the continuity of the mapping R n → F M e ν E , x X λ ∈ Λ T λ ( T − B t x ( θ B ( ˇ ϕ )) χ ) , which, in turn, yields that the mapping R n → E, x T B t x e · X λ ∈ Λ T λ ( T − B t x ( θ B ( ˇ ϕ )) χ ) , is continuous if E is a TMIB and continuous with respect to the weak- ∗ topology on E if E is a DTMIB. Thus, if ϕ ∈ S ( R n ), the mapping (5.11) is continuous if E is aTMIB and continuous with respect to the weak- ∗ topology on E if E is a DTMIB. Nowlet ϕ ∈ ( θ − B W ( F L e ν E , L ω E ))ˇ be arbitrary. Choose a sequence ( ϕ j ) j ∈ N ⊂ S ( R n ) thatconverges to ϕ in ( θ − B W ( F L e ν E , L ω E ))ˇ. Since the mapping W ( F L e ν E , L ω E ) → F L e ν E , ψ ψχ , is continuous, Corollary 3.6 implies that the mappings R n → E, x T x e · X λ ∈ Λ e − πiB t x · λ T λ ( T x ( θ B ( ˇ ϕ j )) χχ ) , j ∈ N , converge pointwise to the mapping (5.11) in E if E is a TMIB and in the weak- ∗ topology of E if E is a DTMIB. This implies the claim. (cid:3) Corollary 5.12.
Let χ, ψ ∈ D ( U ) \{ } be such that ( θ B ( ψ ) , χ ) L = 0 . Then, themappings S χ : E Bd (Λ) → E and R ˇ ψ : E → E Bd (Λ) are continuous and (5.13) R ˇ ψ ◦ S χ = ( θ B ( ψ ) , χ ) L id E Bd (Λ) . In particular, S χ ( E Bd (Λ)) is a complemented subspace of E . ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 21
Proof.
The mapping S χ : E Bd (Λ) → E is continuous by definition of E Bd (Λ) and thecontinuity of the mapping R ˇ ψ : E → E Bd (Λ) has been shown in Proposition 5.10. Theidentity (5.13) follows from a straightforward computation. (cid:3) We end this subsection by giving two examples; further examples shall be discussedin Subsection 5.4 below.
Examples 5.13. ( i ) Let E be a solid TMIB or DTMIB. Fix a bounded open neigh-bourhood of the origin W with W ⊂ U . We define the Banach space E d (Λ) := ( c ∈ C Λ (cid:12)(cid:12)(cid:12) X λ ∈ Λ c λ λ + W ∈ E ) with norm k c k := k P λ ∈ Λ | c λ | λ + W k E . Note that E d (Λ) is solid. We have that E Bd (Λ) = E d (Λ) topologically for all real-valued n × n -matrices B . Hence, in the solid case, ourdefinition coincides with the standard one (cf. [13, Definition 3.4]). Let w be a poly-nomially bounded weight function on R n . For 1 ≤ p ≤ ∞ we define ℓ pw (Λ) as theBanach space consisting of all c ∈ C Λ such that k c k ℓ pw (Λ) := k ( c λ w ( λ )) λ ∈ Λ k ℓ p (Λ) < ∞ .We define c ,w (Λ) as the closed subspace of ℓ ∞ w (Λ) consisting of all c ∈ ℓ ∞ w (Λ) satisfyingthe following property: For every ε > (0) of Λ such thatsup λ ∈ Λ \ Λ (0) | c λ | w ( λ ) ≤ ε . Then, ( L pw ) d (Λ) = ℓ pw (Λ); furthermore, ( C ,w ) Bd (Λ) = c ,w (Λ)for all real-valued n × n -matrices B . A similar statement holds for the weighted mixed-norm spaces considered in Example 3.3( i ).( ii ) Let E be a solid TMIB or DTMIB. We wish to determine ( F E ) d (Λ). We de-fine E ( R n / Λ ⊥ ) as the Banach space consisting of all Λ ⊥ -periodic elements f ∈ E loc with norm k f k E ( R n / Λ ⊥ ) := k f I Λ ⊥ k E . Since E ⊂ L ( R n ), we have that E ( R n / Λ ⊥ ) ⊂ L ( R n / Λ ⊥ ). As customary, we definethe Fourier coefficients of an element f ∈ L ( R n / Λ ⊥ ) as c λ ( f ) = 1vol(Λ ⊥ ) Z I Λ ⊥ f ( x ) e − πiλ · x d x, λ ∈ Λ . We then have:
Proposition 5.14.
Let E be a solid TMIB or DTMIB. Then, (5.14) E ( R n / Λ ⊥ ) → ( F E ) d (Λ) , f ( c λ ( f )) λ ∈ Λ is a topological isomorphism.Proof. Let f ∈ E ( R n / Λ ⊥ ) be arbitrary. Note that ( c λ ( f )) λ ∈ Λ ∈ ℓ ∞ (Λ) ⊂ S ′ d (Λ). Hence, f ( ξ ) = X λ ∈ Λ c λ ( f ) e πiλ · ξ in S ′ ( R n ) . Let χ ∈ D ( U ) \{ } . By Lemma 5.6( ii ), we infer that F − ( S χ (( c λ ( f )) λ ∈ Λ ))( ξ ) = X λ ∈ Λ c λ ( f ) F − ( T λ χ )( ξ ) = F − ( χ )( ξ ) X λ ∈ Λ c λ ( f ) e πiλ · ξ = F − ( χ )( ξ ) f ( ξ ) = X µ ∈ Λ ⊥ F − ( χ )( ξ ) f ( ξ )1 µ + I Λ ⊥ ( ξ ) . For each µ ∈ Λ ⊥ it holds that kF − ( χ ) f µ + I Λ ⊥ k E ≤ ω E ( µ ) k T − µ ( F − ( χ )) f I Λ ⊥ k E ≤ C (1 + | µ | ) τ k T − µ F − ( χ ) k L ∞ ( I Λ ⊥ ) k f k E ( R n / Λ ⊥ ) ≤ C (1 + | µ | ) − n − k f k E ( R n / Λ ⊥ ) , whence the mapping (5.14) is well-defined and continuous. This mapping is injectivebecause for all f ∈ L ( R n / Λ ⊥ ) it holds that f = 0 if and only if c λ ( f ) = 0 forall λ ∈ Λ. Next, let c ∈ ( F E ) d (Λ) be arbitrary. By Proposition 5.5, c ∈ S ′ d (Λ).Hence, f ( ξ ) = P λ ∈ Λ c λ e πiλ · ξ is a well-defined Λ ⊥ -periodic element of S ′ ( R n ). Choose ϕ ∈ S ( R n ) such that F − ( ϕ )( ξ ) = 0 for all ξ ∈ R n . By Lemma 5.6( ii ) and Proposition5.7, we have that E ∋ F − ( S ϕ ( c )) = X λ ∈ Λ c λ F − ( T λ ϕ ) = F − ( ϕ ) f, which implies that f ∈ E ( R n / Λ ⊥ ) and, in view of Proposition 5.7, k f k E ( R n / Λ ⊥ ) = k f I Λ ⊥ k E ≤ k / F − ( ϕ ) k L ∞ ( I Λ ⊥ ) kF − ( ϕ ) f k E ≤ C k c k F E d (Λ) . Clearly, c is equal to the image of f under the mapping (5.14). Therefore, this mappingis surjective and its inverse is continuous. (cid:3) Corollary 5.15.
Let ≤ p ≤ ∞ . Then, ( F L pw ) d (Λ) = ( F L p ) d (Λ) for all polynomiallybounded weight functions w on R n . Convergence properties.
In this subsection we address the following question:
Let χ ∈ D ( U ) \{ } and c ∈ E Bd (Λ) . In which sense does the series P λ ∈ Λ c λ T Bλ χ converge in E ? When E is solid, we can give a quick answer to this question (cf. [13,Proposition 5.2]). Lemma 5.16.
Let E be solid and let χ ∈ D ( U ) \{ } . For each c ∈ E Bd (Λ) the series P λ ∈ Λ c λ T Bλ χ converges unconditionally in E if E is a TMIB and converges uncondi-tionally with respect to the weak- ∗ topology on E if E is a DTMIB.Proof. We only consider the case when E is a TMIB as the case when E is a DTMIB canbe treated similarly. Let ε > ψ ∈ D ( R n ) such that k S χ ( c ) − ψ k E ≤ ε . Let Λ (0) be a finite subset of Λ such that supp ψ ∩ ( S λ ∈ Λ \ Λ (0) ( λ + U )) = ∅ . Forany Λ (0) ⊆ Λ ′ ⊂ Λ, Λ ′ finite, denote by g Λ ′ the characteristic function of the setsupp ψ ∪ ( S λ ∈ Λ ′ ( λ + U )). Since P λ ∈ Λ ′ c λ T Bλ χ = g Λ ′ S χ ( c ) and (1 − g Λ ′ ) ψ = 0, we inferthat (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S χ ( c ) − X λ ∈ Λ ′ c λ T Bλ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E = k (1 − g Λ ′ )( S χ ( c ) − ψ ) k E ≤ k S χ ( c ) − ψ k E ≤ ε. (cid:3) ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 23
However, for general TMIB and DTMIB this question is far more subtle, as thefollowing observation shows.
Proposition 5.17.
Let χ ∈ D (( − , )) \{ } . For ≤ p < ∞ , p = 2 , there existsan element c ∈ ( F L p ) d ( Z ) such that the series P λ ∈ Z c λ T λ χ is not unconditionallyconvergent in F L p ( R ) . For p = 1 there even exists an element c ∈ ( F L ) d ( Z ) suchthat the sequence of symmetric partial sums (cid:16)P | λ |≤ N c λ T λ χ (cid:17) N ∈ N does not converge in F L ( R ) .Proof. In view of Proposition 5.14, this is a consequence of the following two classicalfacts about Fourier series: For 1 ≤ p < ∞ , p = 2, there exists an element in L p ( R / Z )whose Fourier series is not unconditionally convergent in L p ( R / Z ) [21, Exercise 6.5];there exists an element in L ( R / Z ) such that the sequence of symmetric partial sumsof its Fourier series does not converge in L ( R / Z ) [17, Example 4.1.4]. (cid:3) We will now formulate a positive answer to the above question by using the conceptof C´esaro summability.
Definition 5.18.
Let X be a Hausdorff topological vector space and let ( x λ ) λ ∈ Λ ⊂ X .The series P λ ∈ Λ x λ is said to be C´esaro summable to x ∈ X if (recall that Λ = A Λ Z n )lim N →∞ X m ∈ Z n | m j | Let χ ∈ D ( U ) \{ } . For each c ∈ E Bd (Λ) , the series P λ ∈ Λ c λ T Bλ χ isC´esaro summable in E if E is a TMIB and C´esaro summable with respect to the weak- ∗ topology on E if E is a DTMIB. We need some preparation for the proof of Theorem 5.19. A sequence ( k N ) N ∈ Z + ⊂ L ( R n / Z n ) is called an approximate identity on R n / Z n [17, Definition 1.2.15] if( i ) R [ − , ] n k N ( x )d x = 1 for all N ∈ Z + .( ii ) sup N ∈ Z + R [ − , ] n | k N ( x ) | d x < ∞ .( iii ) For all δ > N →∞ Z [ − , ] n \ [ − δ,δ ] n | k N ( x ) | d x = 0 . Set F N ( x ) = X m ∈ Z n | m j | Let ( k N ) N ∈ Z + be an approximate identity on R n / Z n . ( i ) Let X be a Banach space. Suppose that f : [ − , ] n → X is continuous. Then, (5.15) lim N →∞ Z [ − , ] n f ( x ) k N ( x )d x = f (0) , where the above integrals should be interpreted as X -valued Bochner integrals. ( ii ) Let X be a separable Banach space and set X = X ′ . Suppose that f : [ − , ] n → X is continuous with respect to the weak- ∗ topology on X . Then, (5.15) holdswith respect to the weak- ∗ topology on X if the integrals are interpreted as X -valued Pettis integrals with respect to the weak- ∗ topology on X .Proof of Theorem 5.19. Choose ψ ∈ D ( U ) such that ψ = 1 on supp χ . Let c ∈ E Bd (Λ)be arbitrary. For each N ∈ Z + , it holds that X | m j | 0, of the series S χ ( c ) = P λ ∈ Λ c λ T Bλ χ , namely, B αN ( S χ ( c )) = X m ∈ Z n | m |≤ N (cid:18) − | m | N (cid:19) α c A Λ m T BA Λ m χ, N ∈ Z + . Set L αN ( x ) = X m ∈ Z n | m |≤ N (cid:18) − | m | N (cid:19) α e πim · x , N ∈ Z + . Then, ( L αN ) N ∈ Z + is an approximate identity on R n / Z n if α > ( n − / α > ( n − / N →∞ B αN ( S χ ( c )) = S χ ( c )in E if E is a TMIB and with respect to the weak- ∗ topology on E if E is a DTMIB.We denote by c (Λ) the space consisting of all elements of C Λ with only finitelymany non-zero entries. We have the following consequence of Theorem 5.19. Corollary 5.22. Let E be a TMIB. The space c (Λ) is dense in E Bd (Λ) . With the help of Theorem 5.19, we can also describe the bilinear mapping (5.2) fromProposition 5.7. Corollary 5.23. For all c ∈ E Bd (Λ) and ϕ ∈ W ( F L e ν E , L ω E ) , it holds that (5.16) S ϕ ( c ) = X λ ∈ Λ c λ T Bλ ϕ where the series is C´esaro summable in E if E is a TMIB and C´esaro summable withrespect to the weak- ∗ topology on E if E is a DTMIB.Proof. Since S ( R n ) is dense in W ( F L e ν E , L ω E ), Proposition 5.7 yields that (5.16) holdstrue for all c ∈ c (Λ) and ϕ ∈ W ( F L e ν E , L ω E ). Let c ∈ E Bd (Λ) be arbitrary. We define c ( N ) ∈ c (Λ), N ∈ Z + , by c ( N ) A Λ m = ( (cid:16) − | m | N (cid:17) · · · (cid:16) − | m n | N (cid:17) c A Λ m , if | m j | < N, , otherwise.If E is a TMIB, Proposition 5.7 and Theorem 5.19 imply that S ϕ ( c ( N ) ) converges to S ϕ ( c ) in E , which completes the proof in this case. Now suppose that E is a DTMIBwith E = E ′ , where E is a TMIB. Let χ ∈ D ( U ) \{ } be as in the second part ofProposition 5.7. Then, for every g ∈ E there is h ∈ E satisfying * X | m j | In thissubsection, we determine the dual of the discrete space associated to a TMIB and studythe discrete space associated to the completed tensor product of two TMIB. Proposition 5.24. Let E be a TMIB. The strong dual of E Bd (Λ) may be topologicallyidentified with ( E ′ ) − Bd (Λ) via the dual pairing h c ′ , c i = X λ ∈ Λ c ′ λ c λ , c ′ ∈ ( E ′ ) − Bd (Λ) , c ∈ E Bd (Λ) . Furthermore, the series P λ ∈ Λ c ′ λ c λ is C´esaro summable in C .Proof. Let χ, ψ ∈ D ( U ) \{ } be such that R R d χ ( x ) ψ ( x ) dx = 1. Note that X λ ∈ Λ c ′ λ c λ = *X λ ∈ Λ c ′ λ T − Bλ ψ, X λ ∈ Λ c λ T Bλ χ + , c ′ ∈ ( E ′ ) − Bd (Λ) , c ∈ c (Λ) . Hence, Theorem 5.19 implies that the mapping( E ′ ) − Bd (Λ) → ( E Bd (Λ)) ′ b , c ′ c X λ ∈ Λ c ′ λ c λ ! , is well-defined and continuous, and that the series P λ ∈ Λ c ′ λ c λ is C´esaro summable in C . This mapping is clearly injective. We now show that it is also surjective; the resultthen follows from the open mapping theorem. Pick χ ∈ D ( U ) \{ } such that ˇ χ ∈ D ( U )and set ψ = θ − B ( ˇ χ ) ∈ D ( U ) \{ } . Let x ′ ∈ ( E Bd (Λ)) ′ be arbitrary. There is c ′ ∈ C Λ such that h x ′ , c i = X λ ∈ Λ c ′ λ c λ , c ∈ c (Λ) . ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 27 Since the space c (Λ) is dense in E Bd (Λ) (Corollary 5.22), it suffices to show that c ′ ∈ ( E ′ ) − Bd (Λ). Consider the continuous linear mapping R ψ : E → E Bd (Λ) fromProposition 5.10 and denote its transpose by t R ψ . For all ϕ ∈ D ( R n ) it holds that h t R ψ ( x ′ ) , ϕ i = X λ ∈ Λ c ′ λ ϕ ∗ B ψ ( λ )= X λ ∈ Λ c ′ λ Z R n ϕ ( x ) T − Bλ χ ( x )d x = Z R n X λ ∈ Λ c ′ λ T − Bλ χ ! ( x ) ϕ ( x ) dx. Hence, P λ ∈ Λ c ′ λ T − Bλ χ = t R ψ ( x ′ ) ∈ E ′ and, thus, c ′ ∈ ( E ′ ) − Bd (Λ). (cid:3) Our next goal is to show that the completed tensor product of the discrete spacesassociated to two TMIB is canonically isomorphic to the discrete space associated tothe completed tensor product of the two TMIB. Proposition 5.25. Let E j be a TMIB on R n j , let B j be a real-valued n j × n j -matrix,and let Λ j be a lattice in R n j for j = 1 , . Let τ denote either π or ǫ . Then, ( E ) B d (Λ ) b ⊗ τ ( E ) B d (Λ ) is canonically isomorphic to ( E b ⊗ τ E ) B ⊕ B d (Λ × Λ ) .Proof. Set n = n + n , B = B ⊕ B , and Λ = Λ × Λ . Choose a bounded openneighbourhood U j of the origin in R n j such that the families of sets { λ + U j | λ ∈ Λ j } , j = 1 , 2, are pairwise disjoint. Set U = U × U . Choose χ j ∈ D ( U j ) \{ } such thatˇ χ j ∈ D ( U j ), j = 1 , 2, and set χ = χ ⊗ χ ∈ D ( U ) \{ } . Denote by ι : ( E ) B d (Λ ) ⊗ τ ( E ) B d (Λ ) → ( E b ⊗ τ E ) Bd (Λ) the canonical inclusion mapping. We need to show thatthis mapping extends to a topological isomorphism from ( E ) B d (Λ ) b ⊗ τ ( E ) B d (Λ ) onto( E b ⊗ τ E ) Bd (Λ). By the identity S χ ⊗ S χ = S χ ◦ ι and Corollary 5.12, it suffices toshow that the mapping S χ b ⊗ τ S χ : ( E ) B d (Λ ) b ⊗ τ ( E ) B d (Λ ) → E b ⊗ τ E is a topological embedding with range equal to S χ (( E ˆ ⊗ τ E ) Bd (Λ)). The mappings S χ j , j = 1 , 2, are topological embeddings. Hence, by definition of the ǫ -topology,the mapping S χ b ⊗ ǫ S χ is a topological embedding as well (cf. [24, p. 47]). For the π -topology, Corollary 5.12 and [24, Proposition 2.4] imply that also the mapping S χ b ⊗ π S χ is a topological embedding. The identity S χ ⊗ S χ = S χ ◦ ι and the fact that S χ (( E b ⊗ τ E ) Bd (Λ)) is closed in E b ⊗ τ E imply that the range of S χ b ⊗ τ S χ is included in S χ (( E b ⊗ τ E ) Bd (Λ)). As the space c (Λ) = c (Λ ) ⊗ c (Λ ) ⊂ ( E ) B d (Λ ) ⊗ ( E ) B d (Λ )is dense in ( E b ⊗ τ E ) Bd (Λ) (Corollary 5.22) and S χ b ⊗ τ S χ is a topological embedding,we conclude that the range of S χ b ⊗ τ S χ is equal to S χ (( E b ⊗ τ E ) Bd (Λ)). (cid:3) Examples. As explained in the introduction, the 2 n × n -matrix B from Defini-tion 4.1 is the most important for our purposes. In this subsection, we determine thediscrete space associated to various TMIB and DTMIB with respect to B . We startwith the spaces considered in Example 3.3( iii ). Let Λ be a lattice in R n and let B be a real-valued n × n matrix. For c ∈ C Λ we set θ B ( c ) = ( c λ e πiBλ · λ ) λ ∈ Λ . Given a Banach space X ⊂ C Λ , we define the Banach space θ B X := { c ∈ C Λ | θ − B ( c ) ∈ X } with norm k c k θ B X := k θ − B ( c ) k X . Proposition 5.26. Let E be a TMIB on R n and let w be a polynomially boundedweight function on R n . Let Λ and Λ be two lattices in R n . ( i ) It holds that ( L pw ( R nξ ; E x )) B d (Λ ,x × Λ ,ξ ) = ℓ pw (Λ ; E d (Λ )) , ≤ p < ∞ , ( C ,w ( R nξ ; E x )) B d (Λ ,x × Λ ,ξ ) = c ,w (Λ ; E d (Λ )) , topologically. ( ii ) Suppose that ν E = 1 . Then, ( L pw ( R nx ; E ξ )) B d (Λ ,x × Λ ,ξ ) = θ − B ℓ pw (Λ ; E d (Λ )) , ≤ p < ∞ , ( C ,w ( R nx ; E ξ )) B d (Λ ,x × Λ ,ξ ) = θ − B c ,w (Λ ; E d (Λ )) , topologically.Proof. We only show the statements for L pw as the proofs for C ,w are similar. SetΛ = Λ × Λ . Choose a bounded open neighbourhood U of the origin such that thefamilies of sets { λ j + U | λ j ∈ Λ j } , j = 1 , 2, are pairwise disjoint. Fix χ j ∈ D ( U ) with χ j (0) = 0, j = 1 , 2, and set χ = χ ⊗ χ ∈ D ( U × U ) \{ } .( i ) Since c (Λ) is dense in both ( L pw ( R nξ ; E x )) B d (Λ × Λ ) and ℓ pw (Λ ; E d (Λ )) (Corollary5.22), it suffices to show that these spaces induce the same topology on c (Λ). For all c = ( c λ ,λ ) ( λ ,λ ) ∈ Λ ∈ c (Λ) it holds that k S χ ( c ) k pL pw ( R nξ ; E x ) = Z R n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ ∈ Λ T λ χ ( ξ ) X λ ∈ Λ c λ ,λ e − πiλ ( ξ − λ ) T λ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pE w ( ξ ) p d ξ = Z R n X λ ∈ Λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ ∈ Λ c λ ,λ e − πiλ ( ξ − λ ) T λ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pE | T λ χ ( ξ ) | p w ( ξ ) p d ξ = Z R n X λ ∈ Λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M − ( ξ − λ ) X λ ∈ Λ c λ ,λ T λ M ξ − λ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pE | T λ χ ( ξ ) | p w ( ξ ) p d ξ. Since the set { M η χ | η ∈ U } is bounded in D ( U ) and χ = 0, Lemma 5.4 implies thatthere is C > η ∈ U and c ∈ c (Λ), C − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ ∈ Λ c λ ,λ T λ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ ∈ Λ c λ ,λ T λ M η χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ ∈ Λ c λ ,λ T λ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E . Hence, k S χ ( c ) k pL pw ( R nξ ; E x ) ≤ X λ ∈ Λ Z R n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ ∈ Λ c λ ,λ T λ M ξ − λ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pE | T λ ( χ ν E )( ξ ) | p w ( ξ ) p d ξ ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 29 ≤ C p X λ ∈ Λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ ∈ Λ c λ ,λ T λ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pE Z U | χ ( ξ ) | p ν E ( ξ ) p w ( ξ + λ ) p d ξ ≤ C ′ X λ ∈ Λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ ∈ Λ c λ ,λ T λ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pE w ( λ ) p = C ′ k c k pℓ pw (Λ ; E d (Λ )) for all c ∈ c (Λ). Next, choose an open neighbourhood V of 0 such that inf ξ ∈ V | χ ( ξ ) | > 0. Then, k S χ ( c ) k pL pw ( R nξ ; E x ) ≥ X λ ∈ Λ Z R n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ ∈ Λ c λ ,λ T λ M ξ − λ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pE | T λ ( χ / ˇ ν E )( ξ ) | p w ( ξ ) p d ξ ≥ C − p X λ ∈ Λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ ∈ Λ c λ ,λ T λ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pE Z V | χ ( ξ ) | p ˇ ν E ( ξ ) − p w ( ξ + λ ) p d ξ ≥ C ′− X λ ∈ Λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ ∈ Λ c λ ,λ T λ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pE w ( λ ) p = C ′− k c k pℓ pw (Λ ; E d (Λ )) for all c ∈ c (Λ). This shows the result.( ii ) As in part ( i ), it suffices to show that the spaces ( L pw ( R nx ; E ξ )) B d (Λ ,x × Λ ,ξ ) and θ − B ℓ pw (Λ ; E d (Λ )) induce the same topology on c (Λ). Let c ∈ c (Λ) be arbitraryand set ˜ c = θ B ( c ). Then, S χ ( c ) = X λ ∈ Λ T λ χ ⊗ M − λ X λ ∈ Λ ˜ c λ ,λ T λ χ ! and thus k S χ ( c ) k pL pw ( R nx ; E ξ ) = X λ ∈ Λ k T λ χ k pL pw (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M − λ X λ ∈ Λ ˜ c λ ,λ T λ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pE . We infer that C − k ˜ c k pℓ pw (Λ ; E d (Λ )) ≤ k S χ ( c ) k L pw ( R nx ; E ξ ) ≤ C k ˜ c k pℓ pw (Λ ; E d (Λ )) , from which the result follows. (cid:3) Remark . If E , w , Λ and Λ are as in Proposition 5.26( i ), the exact same argumentas in its proof shows that( L pw ( R nξ ; E x )) − B d (Λ ,x × Λ ,ξ ) = ℓ pw (Λ ; E d (Λ )) , ≤ p < ∞ , topologically. Proposition 5.28. Let E be a DTMIB with the Radon-Nikod´ym property and let w be a polynomially bounded weight function on R n . Let Λ and Λ be two lattices in R n . ( i ) It holds that ( L pw ( R nξ ; E x )) B d (Λ ,x × Λ ,ξ ) = ℓ pw (Λ ; E d (Λ )) , < p ≤ ∞ , topologically. ( ii ) Suppose that ν E = 1 . Then, ( L pw ( R nx ; E ξ )) B d (Λ ,x × Λ ,ξ ) = θ − B ℓ pw (Λ ; E d (Λ )) , < p ≤ ∞ , topologically.Proof. We only show ( i ) as ( ii ) can be treated similarly. Suppose that E = E ′ , where E is a TMIB. Let q be the H¨older conjugate index to p . As E satisfies the Radon-Nikod´ym property, we have that L pw ( R n ; E ) = ( L q /w ( R n ; E )) ′ . Proposition 5.24 yields that E d (Λ ) = (( E ) d (Λ )) ′ . Since, by Corollary 5.12, E d (Λ )also satisfies the Radon-Nikod´ym property, we have that ℓ pw (Λ ; E d (Λ )) = ( ℓ q /w (Λ ; ( E ) d (Λ ))) ′ . The result now follows from Proposition 5.24 and 5.27. (cid:3) Next, we discuss completed tensor products of two TMIB on R n . Note that B isnot the direct sum of two n × n -matrices and therefore this matrix is not covered byProposition 5.25. It would be interesting to determine the discrete space associated tothe completed tensor product of two TMIB with respect to B but even for most ofthe L p -spaces we do not know how to do this: Problem 5.29. Let Λ and Λ be lattices in R n . Let τ denote either π or ǫ . Give anexplicit description of ( L p b ⊗ τ L p ) B d (Λ × Λ ) for 1 ≤ p , p < ∞ .For τ = π and p = 1 we have the following (trivial) answer to Problem 5.29 (cf. [24,Section 2.3]):( L b ⊗ π L p ) B d (Λ × Λ ) = ( L ( L p )) B d (Λ × Λ ) = ℓ (Λ ; ℓ p (Λ )) , where the second equality follows from Example 5.13( i ).We now provide an answer to Problem 5.29 for p = 2 and p varying in a certainrange. In fact, we are able to show the following more general result. Proposition 5.30. Let Λ and Λ be two lattices in R n , let w and w be two polyno-mially bounded weight functions on R n , and let ≤ p , p < ∞ . Then, ( i ) ( L p w b ⊗ π F L p w ) B d (Λ × Λ ) = θ − B ℓ w w (Λ ; ( F L p ) d (Λ )) if p − + p − ≥ , ( ii ) ( L p w b ⊗ ǫ F L p w ) B d (Λ × Λ ) = θ − B c ,w w (Λ ; ( F L p ) d (Λ )) if p − + p − ≤ ,topologically.Proof. Set Λ = Λ × Λ . Let U ⊂ R n be a bounded open neighbourhood of the ori-gin such that the families { λ j + U | λ j ∈ Λ j } , j = 1 , 2, are pairwise disjoint. Choose A j , κ j > w j ( x + y ) ≤ A j w j ( x )(1 + | y | ) κ j , j = 1 , 2, for all x, y ∈ R n . Wewrite q j for the H¨older conjugate index to p j , j = 1 , 2. By the closed graph theorem,it suffices to show that the identities in ( i ) and ( ii ) hold algebraically. ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 31 ( i ) We first show that θ − B ℓ w w (Λ ; ( F L p ) d (Λ )) ⊆ ( L p w b ⊗ π F L p w ) B d (Λ). Pick χ , χ ∈D ( U ) \{ } and set χ = χ ⊗ χ ∈ D ( U × U ) \{ } . Let c ∈ θ − B ℓ w w (Λ ; ( F L p ) d (Λ ))be arbitrary and set ˜ c = θ B ( c ) ∈ ℓ w w (Λ ; ( F L p ) d (Λ )). We have that(5.17) S χ ( c ) = X λ ∈ Λ T λ χ ⊗ M − λ X λ ∈ Λ ˜ c λ ,λ T λ χ ! . Corollary 5.15 implies that, for λ ∈ Λ fixed, (˜ c λ ,λ ) λ ∈ Λ ∈ ( F L p w ) d (Λ ) and that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M − λ X λ ∈ Λ ˜ c λ ,λ T λ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F L p w ≤ A w ( λ ) k (˜ c λ ,λ ) λ ∈ Λ k ( F L p | · | ) κ ) d (Λ ) ≤ Cw ( λ ) k (˜ c λ ,λ ) λ ∈ Λ k ( F L p ) d (Λ ) . We obtain that the series in the right-hand side of (5.17) (over Λ ) is absolutelysummable in L p w b ⊗ π F L p w . This shows the desired inclusion. Next, we prove that( L p w b ⊗ π F L p w ) B d (Λ) ⊆ θ − B ℓ w w (Λ ; ( F L p ) d (Λ )). Let r > − r, r ] n ⊂ U . Pick ψ ∈ D [ − r,r ] n \{ } and set ψ ( x, ξ ) = e − πix · ξ ψ ( x ) ψ ( ξ ) and ψ = ( ψ ∗ B ψ )ˇ . We choose ψ so that ψ is not the zero function. Then, ψ ∈ D [ − r,r ] n \{ } and ψ ∈D ( U × U ) \{ } . Furthermore, choose χ ∈ D ( U × U ) such that ( θ B ( ψ ) , χ ) L = 1.Corollary 5.12 implies that(5.18) c = R ˇ ψ ( S χ ( c )) ∈ R ˇ ψ ( L p w b ⊗ π F L p w ) , c ∈ ( L p w b ⊗ π F L p w ) B d (Λ) . We claim that(5.19) F ∗ B ψ ∈ L w w ( F L p ) , F ∈ L p w b ⊗ π F L p w . Before we prove (5.19), let us show how it entails the result. For all F ∈ L p w b ⊗ π F L p w ,we have that R ˇ ψ ( F ) = ( F ∗ B ( ψ ∗ B ψ )( λ )) λ ∈ Λ = (( F ∗ B ψ ) ∗ B ψ ( λ )) λ ∈ Λ = R ψ ( F ∗ B ψ ) . Hence, in view of (5.18) and (5.19), the desired inclusion follows from Proposition 5.10and Proposition 5.26( ii ). It remains to prove (5.19). Let F ∈ L p w b ⊗ π F L p w be arbitrary.Then,(5.20) F = ∞ X j =0 a j f j ⊗ g j , where f j , g j ∈ S ( R n ) are such that ( f j ) j ∈ N is bounded in L p w and ( g j ) j ∈ N is boundedin F L p w , and a j ∈ C are such that P ∞ j =0 | a j | < ∞ (cf. [24, Proposition 2.8]). For all f, g ∈ S ( R n ) it holds that( f ⊗ g ) ∗ B ψ ( t, η ) = Z Z R n f ( x ) g ( ξ ) ψ ( t − x ) ψ ( η − ξ ) e − πit · ( η − ξ ) d x d ξ (5.21) = f ∗ ψ ( t ) g ∗ ( M − t ψ )( η ) . We estimate as follows k ( f ⊗ g ) ∗ B ψ k L w w ( F L p ) = Z R n | f ∗ ψ ( t ) | w ( t ) w ( t ) k ( F − g )( T t F − ψ ) k L p d t ≤ A k f ∗ ψ k L q w (cid:18)Z Z R n |F − g ( ξ ) | p w ( ξ ) p |F − ψ ( ξ − t ) | p (1 + | t − ξ | ) κ p d t d ξ (cid:19) /p = A k f ∗ ψ k L q w k g k F L p w k ψ k F L p | · | ) κ ≤ C k f k L p w k g k F L p w , where the last inequality follows from Young’s inequality (note that q ≥ p ). Therepresentation (5.20) and the above estimate yield that F ∗ B ψ ∈ L w w ( F L p ).( ii ) The assumption on p and p implies that 1 < p , p < ∞ and thus also 1 Denote these sums by I ′ and I ′′ , respectively. We estimate I ′ as follows I ′ ≤ k χ k L p X λ ∈ Λ ′ \ Λ (0)1 k f k L q ( λ + U ) k ( M − λ f ) ∗ ˇ˜ χ k F L q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ ∈ Λ ˜ c λ ,λ T λ ˜ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F L p ≤ ε k χ k L p X λ ∈ Λ ′ \ Λ (0)1 k f k L q ( λ + U ) w ( λ ) · kF − f T − λ b ˜ χ k L q w ( λ ) ≤ ε k χ k L p X λ ∈ Λ k f k p L q ( λ + U ) w ( λ ) p ! /p X λ ∈ Λ kF − f T − λ b ˜ χ k q L q w ( λ ) q ! /q . Since q ≤ p , we infer that X λ ∈ Λ k f k p L q ( λ + U ) w ( λ ) p ! /p ≤ A A X λ ∈ Λ k f k p L q /w ( λ + U ) ! /p ≤ A A k f k L q /w ≤ A A . Furthermore, X λ ∈ Λ kF − f T − λ b ˜ χ k q L q w ( λ ) q ≤ A q Z R n |F − f ( ξ ) | q ˇ w ( ξ ) q X λ ∈ Λ | b ˜ χ ( ξ + λ ) | q (1 + | ξ + λ | ) κ q d ξ ≤ A q A q k f k q F L q / ˇ w ≤ A q A q . Plugging these bounds into the above estimate for I ′ , we deduce that I ′ ≤ ε/ 2. Anal-ogously, we find that I ′′ ≤ ε/ 2. Hence,sup f ∈ K sup f ∈ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* f ⊗ f , X λ ∈ Λ ′ T λ χ ⊗ g λ − X λ ∈ Λ ′′ T λ χ ⊗ g λ +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε, from which the statement and therefore also the desired inclusion follows. Finally,we prove that ( L p w b ⊗ ǫ F L p w ) B d (Λ) ⊆ θ − B c ,w w (Λ ; ( F L p ) d (Λ )). Let ψ , ψ and ψ be as in the second part of the proof of part ( i ). Pick ˜ χ , χ ∈ D ( U ) \{ } such that χ = ˜ χ ∗ ˜ χ ∗ ˜ χ ∈ D ( U ) \{ } and set χ = χ ⊗ χ ∈ D ( U × U ) \{ } . Choose ˜ χ and χ such that ( θ B ( ψ ) , χ ) L = 1. Corollary 5.12 implies that(5.22) c = R ˇ ψ ( S χ ( c )) , c ∈ ( L p w b ⊗ ǫ F L p w ) B d (Λ) . Arguing as in the proof of part ( i ), we see that it suffices to show that(5.23) S χ ( c ) ∗ B ψ ∈ C ,w w ( F L p ) , c ∈ ( L p w b ⊗ ǫ F L p w ) B d (Λ) . Without loss of generality, we may assume that w and w are continuous. Then, (cf.[24, Section 3.1])(5.24) C ,w ( F L p w ) = C ,w b ⊗ ǫ F L p w . Given a continuous polynomially bounded weight function w on R n , we denote by J w the isometrical isomorphism J w : C → C ,w , ϕ ϕ/w . Then, t J w : ( C ,w ) ′ → M is an isometrical isomorphism. Set ˜ χ = ˜ χ ⊗ χ ∈ D ( U × U ) \{ } . Let c ∈ c (Λ) bearbitrary. For all f ∈ ( C ,w ) ′ and f ∈ ( F L p w ) ′ , it holds that(5.25) h f ⊗ f , S χ ( c ) i = h ( f ∗ ˇ˜ χ ∗ ˇ˜ χ ) ⊗ f , S ˜ χ ( c ) i . By Young’s inequality, we infer that k ( f ∗ ˇ˜ χ ) ∗ ˇ˜ χ k L q /w ≤ A k ˇ˜ χ k L q | · | ) κ k f ∗ ˇ˜ χ k L /w ≤ A k ˇ˜ χ k L | · | ) κ k ˇ˜ χ k L q | · | ) κ k t J w f k M = A k ˇ˜ χ k L | · | ) κ k ˇ˜ χ k L q | · | ) κ k f k ( C ,w ) ′ . Hence, in view of (5.24), (5.25) and the fact that c (Λ) is dense in ( L p w b ⊗ ǫ F L p w ) B d (Λ)(Corollary 5.22), we deduce that the mapping( L p w b ⊗ ǫ F L p w ) B d (Λ) → C ,w ( F L p w ) , c S χ ( c ) , is well-defined and continuous. Consequently, to prove (5.23), it suffices to show thatthe mapping C ,w ( F L p w ) → C ,w w ( F L p ) , F F ∗ B ψ , is well-defined and continuous. Let F ∈ S ( R n ) be arbitrary. Note that (cf. (5.21))id b ⊗F − ( F ∗ B ψ )( t, η ) = Z Z R n F ( x, ξ ) ψ ( t − x ) b ψ ( t − η ) e πiξ · η d x d ξ = b ψ ( t − η ) Z R n id b ⊗F − ( F )( x, η ) ψ ( t − x )d x. We infer that k F ∗ B ψ k L ∞ w w ( F L p ) ≤ sup t ∈ R n w ( t ) w ( t ) (cid:18)Z R n | b ψ ( t − η ) | p (cid:18)Z R n | id b ⊗F − ( F )( x, η ) || ψ ( t − x ) | d x (cid:19) p d η (cid:19) /p ≤ A A k ψ k L ∞ (1+ | · | ) κ k b ψ k L ∞ (1+ | · | ) κ × sup t ∈ R n (cid:18)Z R n (cid:18)Z t − U | id b ⊗F − ( F )( x, η ) | w ( x )d x (cid:19) p w ( η ) p d η (cid:19) /p ≤ A A | U | /q k ψ k L ∞ (1+ | · | ) κ k b ψ k L ∞ (1+ | · | ) κ × sup t ∈ R n (cid:18)Z R n Z t − U | id b ⊗F − ( F )( x, η ) | p w ( x ) p w ( η ) p d x d η (cid:19) /p ≤ A A | U |k ψ k L ∞ (1+ | · | ) κ k b ψ k L ∞ (1+ | · | ) κ k id b ⊗F − ( F ) k L ∞ w ( L p w ) = A A | U |k ψ k L ∞ (1+ | · | ) κ k b ψ k L ∞ (1+ | · | ) κ k F k L ∞ w ( F L p w ) . The statement now follows from the density of S ( R n ) in C ,w ( F L p w ). (cid:3) Corollary 5.31. Let Λ and Λ be two lattices in R n and let w be a polynomiallybounded weight function on R n . Then, ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 35 ( i ) ( L pw b ⊗ π L ) B d (Λ × Λ ) = ℓ w (Λ ; ℓ (Λ )) if ≤ p ≤ , ( ii ) ( L pw b ⊗ ǫ L ) B d (Λ × Λ ) = c ,w (Λ ; ℓ (Λ )) if ≤ p < ∞ ,topologically. Gabor frame characterisations of modulation spaces defined viaTMIB and DTMIB The short-time Fourier transform and Gabor frames on S ′ ( R n ) . We startwith a brief discussion of the short-time Fourier transform (STFT) and Gabor frames on L ( R n ); we refer to the book [19] for more information. Recall that for z = ( x, ξ ) ∈ R n we write π ( z ) = M ξ T x . The STFT of f ∈ L ( R n ) with respect to a window ψ ∈ L ( R n )is defined as V ψ f ( x, ξ ) := ( f, π ( x, ξ ) ψ ) L = Z R n f ( t ) ψ ( t − x ) e − πiξ · t d t. Then, V ψ f ∈ L ( R n ) ∩ C ( R n ) and the following orthogonality relation holds(6.1) ( V ψ f, V γ ϕ ) L = ( f, ϕ ) L ( γ, ψ ) L , where also ϕ, γ ∈ L ( R n ). Furthermore, it holds that(6.2) V ψ ( π ( x, ξ ) f ) = T σ ( x,ξ ) V ψ f. Let ψ, γ ∈ L ( R n ) be such that ( γ, ψ ) L = 0. The equations (6.1) and (6.2) imply thereproducing formula(6.3) V ϕ f = 1( γ, ψ ) L V ψ f V ϕ γ, where f, ϕ ∈ L ( R n ).Next, we discuss Gabor frames. Fix a lattice Λ in R n . Let ψ ∈ L ( R n ) and supposethat the analysis operator C ψ : L ( R n ) → ℓ (Λ) , f ( V ψ f ( λ )) λ ∈ Λ , is continuous; this is e.g. the case if ψ ∈ W ( L ∞ , L ) [19, Corollary 6.2.3]. The adjointoperator of C ψ , called the synthesis operator , is given by D ψ : ℓ (Λ) → L ( R n ) , c X λ ∈ Λ c λ π ( λ ) ψ, and the series P λ ∈ Λ c λ π ( λ ) ψ converges unconditionally in L ( R n ). Let ψ, γ ∈ L ( R n )be windows such that C ψ and C γ are continuous. We define S ψ,γ := D γ ◦ C ψ : L ( R n ) → L ( R n )and call ( ψ, γ ) a pair of dual windows on Λ if S ψ,γ = id L ( R n ) . In such a case, also S γ,ψ = id L ( R n ) and thus f = X λ ∈ Λ V ψ f ( λ ) π ( λ ) γ = X λ ∈ Λ V γ f ( λ ) π ( λ ) ψ, f ∈ L ( R n ) , where both series converge unconditionally in L ( R n ). Given a window ψ ∈ L ( R n ), the set of time-frequency shifts G (Λ , ψ ) := { π ( λ ) ψ | λ ∈ Λ } is called a Gabor frame if there are A, B > A k f k L ≤ k ( V ψ f ( λ )) λ ∈ Λ k ℓ (Λ) ≤ B k f k L , f ∈ L ( R n ) . Then, S = S ψ,ψ is a bounded positive invertible linear operator on L ( R n ). Set γ ◦ = S − ψ ∈ L ( R n ). Since S and π commute on Λ, ( ψ, γ ◦ ) is a pair of dual windows on Λ.We call γ ◦ the canonical dual window on Λ of ψ .We now discuss the STFT and Gabor frames on S ′ ( R n ) (cf. [19, Section 11.2] and[22]). Let ψ ∈ S ( R n ). Then, the mapping V ψ : S ( R n ) → S ( R n ) is continuous. TheSTFT of f ∈ S ′ ( R n ) with respect to ψ is defined as(6.4) V ψ f ( x, ξ ) := h f, π ( x, − ξ ) ψ i , ( x, ξ ) ∈ R n . Then, V ψ f ∈ C ( R n ) and k V ψ f k L ∞ (1+ | · | ) − N < ∞ for some N ∈ N . If A ⊂ S ′ ( R n )is bounded, then the previous estimate holds uniformly for f ∈ A . Since S ′ ( R n ) isbornological, this implies that the mapping V ψ : S ′ ( R n ) → S ′ ( R n ) is continuous. Let ψ, γ ∈ S ( R n ) be such that ( γ, ψ ) L = 0. As L ( R n ) is dense in S ′ ( R n ), (6.1) impliesthat(6.5) h f, ϕ i = 1( γ, ψ ) L Z Z R n V ψ f ( x, ξ ) V γ ϕ ( x, − ξ )d x d ξ, ϕ ∈ S ( R n ) , whereas (6.3) yields that(6.6) V ϕ f = 1( γ, ψ ) L V ψ f V ϕ γ, f ∈ S ′ ( R n ) , ϕ ∈ S ( R n ) . Clearly, (6.2) remains true for f ∈ S ′ ( R n ) and ψ ∈ S ( R n ).Finally, we discuss Gabor frames on S ′ ( R n ). Let ψ ∈ S ( R n ). The mappings C ψ : S ′ ( R n ) → S ′ d (Λ) , f ( V ψ f ( λ )) λ ∈ Λ , and D ψ : S ′ d (Λ) → S ′ ( R n ) , c X λ ∈ Λ c λ π ( λ ) ψ, are well-defined and continuous, and the series P λ ∈ Λ c λ π ( λ ) ψ is absolutely summablein S ′ ( R n ). Let ψ, γ ∈ S ( R n ) be such that ( ψ, γ ) is a pair of dual windows on Λ. Then, f = X λ ∈ Λ V ψ f ( λ ) π ( λ ) γ = X λ ∈ Λ V γ f ( λ ) π ( λ ) ψ, f ∈ S ′ ( R n ) , where both series are absolutely summable in S ′ ( R n ). ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 37 Continuity of the Gabor frame operators on modulation spaces associ-ated to TMIB and DTMIB. Fix a TMIB or a DTMIB F on R n . We start bydefining the modulation space associated to F [10]. Definition 6.1. Let ψ ∈ S ( R n ) \{ } . We define the modulation space associated to F as M F := { f ∈ S ′ ( R n ) | V ψ f ∈ F } and endow it with the norm k f k M F := k V ψ f k F .We sometimes employ the alternative notation M [ F ] for M F . The space M F isa Banach space whose definition is independent of the window ψ ∈ S ( R n ) \{ } anddifferent non-zero windows induce equivalent norms on M F [10, Corollary 4.5 andCorollary 4.6]. Furthermore, if F is a TMIB, then M F is a TMIB [10, Theorem4.8( i )]. We define ˇ F := { f ∈ S ′ ( R n ) | ˇ f ( x, ξ ) := f ( x, − ξ ) ∈ F } and endow it with the norm k f k ˇ F := k ˇ f k F . It is clear that ˇ F is again a TMIB(DTMIB). The following duality result holds. Proposition 6.2. [10, Theorem 4.8( iii )] Suppose that F is a TMIB. Then, M F ′ =( M ˇ F ) ′ . Moreover, for ψ, γ ∈ S ( R n ) with ( γ, ψ ) L = 0 , it holds that (cf. (6.5) ) h f, g i = 1( γ, ψ ) L h V ψ f ( x, ξ ) , V γ g ( x, − ξ ) i , f ∈ M F ′ , g ∈ M ˇ F . Consequently, M F is a DTMIB if F is so. Remark . The identity (6.2) implies that k π ( x, ξ ) k L ( M F ) ≤ ρ B F ( x, ξ ) , ( x, ξ ) ∈ R n . Hence, [19, Theorem 12.1.9] gives the continuous inclusion(6.7) M , ρ B F ⊆ M F , which improves [10, Corollary 4.11].For the main result of this article we need to enlarge the class of windows for theSTFT of the elements of M F in such a way that its range consists of continuousfunctions on R n . Given a Banach space X ⊂ S ′ ( R n ), we define the Banach space X := { f ∈ S ′ ( R n ) | f ∈ X } with norm k f k X := k f k X . Assume that F is a TMIB. For f ∈ M F and ψ ∈ M [( F ′ )ˇ ] we define V ψ f ( x, ξ ) := M F h f, π ( x, − ξ ) ψ i M ( F ′ )ˇ2 . Similarly, for f ∈ M F ′ and ψ ∈ M [ ˇ F ] we define V ψ f ( x, ξ ) := M F ′ h f, π ( x, − ξ ) ψ i M ˇ F . Obviously, these definitions coincide with the one given in (6.4) if ψ ∈ S ( R n ). Since( T B ( x, − ξ ) G )ˇ = T − B ( x,ξ ) ˇ G for all G ∈ S ′ ( R n ), Proposition 6.2 together with (6.2) implythat the sesquilinear mappings M F × M [( F ′ )ˇ ] → C / ˇ ρ B F ( R n ) , ( f, ψ ) V ψ f and M F ′ × M [ ˇ F ] → C / ˇ ρ B F ′ ( R n ) , ( f, ψ ) V ψ f are well-defined and continuous. Now suppose again that F is either a TMIB or aDTMIB. Since V ψ f = ( V ψ f )ˇ , f ∈ S ′ ( R n ) , ψ ∈ S ( R n ) , (6.8) W ( F L e ν F , L ω F ) = W ( F L e ν F , L ω F ) , (6.9)Corollary 5.8 implies that M [ W ( F L e ν F , L ω F )] ⊂ M [( F ′ )ˇ ] continuously if F is a TMIBand M [ W ( F L e ν F , L ω F )] ⊂ M [( F )ˇ ] continuously if F is a DTMIB with F = F ′ , where F is a TMIB. Hence, the sesquilinear mapping(6.10) M F × M [ W ( F L e ν F , L ω F )] → C / ˇ ρ B F ( R n ) , ( f, ψ ) V ψ f is well-defined and continuous. Remark . Although we will not need this, we would like to point out that it is alsopossible to enlarge the class of windows for the STFT of the elements in M F in sucha way that its range is in F : Proposition . The sesquilinear mapping M F × S ( R n ) → F , ( f, ψ ) V ψ f , uniquelyextends to a continuous sesquilinear mapping M F × M , ρ Bt F → F , ( f, ψ ) V ψ f .Proof. For all G ∈ S ′ ( R n ) and Φ ∈ S ( R n ), it holds that(6.11) G Z Z R n Φ( x, ξ ) T B t ( x,ξ ) G d x d ξ, where the integral should be interpreted as an S ′ ( R n )-valued Pettis integral with respectto the weak- ∗ topology on S ′ ( R n ). If G ∈ F , then the above integral exists as an F -valued Bochner integral if F is a TMIB and as an F -valued Pettis integral if F is aDTMIB. Consequently, G ∈ F and(6.12) k G k F ≤ k G k F k Φ k L ρBt F , G ∈ F, Φ ∈ S ( R n ) . Now fix γ ∈ S ( R n ) with k γ k L = 1. Let f ∈ M F and ψ ∈ S ( R n ) be arbitrary. Notethat V ψ γ = ( θ − B ( V γ ψ ))ˇ. Hence, the reproducing formula (6.6) and (6.12) yield that k V ψ f k F = k V γ f V ψ γ k F ≤ k V γ f k F k V ψ γ k L ρBt F = k V γ f k F k V γ ψ k L ρBt F , whence the result follows from the density of S ( R n ) in M , ρ Bt F . (cid:3) ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 39 Fix a lattice Λ in R n and a bounded open neighbourhood U of the origin in R n such that the family of sets { λ + U | λ ∈ Λ } is pairwise disjoint. We are ready toestablish the continuity of the analysis and synthesis operators on M F . Recall that F σd (Λ) = F B d (Λ). Theorem 6.6. ( i ) Let ψ ∈ M [ W ( F L e ν F , L ω F )] . The mapping C ψ : M F → F σd (Λ) is well-definedand continuous. ( ii ) Let ψ ∈ M [ W ( F L e ν F , L ω F )] . For each c ∈ F σd (Λ) the series P λ ∈ Λ c λ π ( λ ) ψ isC´esaro summable in M F if F is a TMIB and C´esaro summable with respect tothe weak- ∗ topology on M F if F is a DTMIB (cf. Proposition 6.2). Further-more, the mapping D ψ : F σd (Λ) → M F is well-defined and continuous.Proof. ( i ) Let f ∈ M F be arbitrary. As V ψ f is continuous, we can evaluate it at λ ∈ Λ.Pick γ ∈ S ( R n ) such that k γ k L = 1. Note that, by (6.9), V ψ γ = ( θ − B ( V γ ψ ))ˇ ∈ ( θ − B W ( F L e ν F , L ω F ))ˇ . Since S ( R n ) is dense in M [ W ( F L e ν F , L ω F )], the reproducing formula (6.6) and thecontinuity of the mappings in (5.8) and (6.10) imply that V ψ f = V γ f V ψ γ . Hence, theresult follows from Proposition 5.10.( ii ) In view of (6.2), this is a consequence of Proposition 5.7 and Corollary 5.23 (andProposition 6.2 if F is a DTMIB). (cid:3) Corollary 6.7. Let ψ ∈ M [ W ( F L e ν F , L ω F )] ∩ L and γ ∈ M [ W ( F L e ν F , L ω F )] ∩ L besuch that ( ψ, γ ) is a pair of dual windows on Λ . Then, (6.13) f = X λ ∈ Λ V ψ f ( λ ) π ( λ ) γ, f ∈ M F , where the series is C´esaro summable in M F if F is a TMIB and C´esaro summablewith respect to the weak- ∗ topology on M F if F is a DTMIB. Furthermore, there are A, B > such that A k f k M F ≤ k ( V ψ f ( λ )) λ ∈ Λ k F σd (Λ) ≤ B k f k M F , f ∈ M F . Proof. Note that D γ ◦ C ψ restricts to the identity on S ( R n ). Hence, if F is a TMIB,the result follows from the density of S ( R n ) in M F and Theorem 6.6. Assume nowthat F is a DTMIB. Theorem 6.6 and (6.8) imply that for all χ ∈ S ( R n ) and f ∈ M F h D γ C ψ ( f ) , χ i = lim N →∞ * f , X | m j | Remark . Let ω and ν be submultiplicative polynomially bounded weight functionson R n and set X = W ( F L ν , L ω ). Then, ω X ( x, ξ ) ≤ Cω ( x, ξ ) and ν X ( x, ξ ) ≤ Cν ( x, ξ ).Hence, (4.2) and (6.7) gives the inclusions(6.14) M , σ F ⊆ M [ W ( F L e ν F , L ω F )] and M , σ F ⊆ M [ W ( F L e ν F , L ω F )] , where σ F ( x, ξ ) = ω F ( x, ξ )˜ ν F (0 , x ). If ν F (0 , · ) = 1, the above inequality and theinclusion W ( F L , L ω F ) ⊆ W ( L ∞ , L ω F ) ⊂ L ω F imply that(6.15) M , ω F = M [ W ( F L , L ω F )] and M , ω F = M [ W ( F L , L ω F )] . By (6.14), we can take ψ ∈ M , σ F in Theorem 6.6( i ) and ψ ∈ M , σ F in Theorem 6.6( ii );a similar statement holds for Corollary 6.7. As mentioned in the introduction, if F is solid, Theorem 6.6 and Corollary 6.7 are known to hold true for the window class M , { ω F , ˇ ω F } [13, 18, 19]. The equalities in (6.15) imply that this remains valid for thelarger class of TMIB and DTMIB F for which ν F (0 , · ) = 1; e.g F = E b ⊗ τ E , τ = π or ǫ , where E is a TMIB on R n and E is a solid TMIB on R n , satisfy ν F (0 , · ) = 1. Remark . For each ϕ ∈ W ( L ∞ , L ) \{ } , the system G ( ϕ, a Z n × b Z n ) is a Gaborframe for a, b > ϕ ( x ) = 2 n/ e − πx · x is theGaussian, G ( ϕ, a Z n × b Z n ) is a Gabor frame if and only if ab < ϕ ∈ S ( R n ) and G ( ϕ, a Z n × b Z n ) is a Gabor frame, then the canonical dualwindow γ = S − ϕ on a Z n × b Z n also belongs to S ( R n ) [19, Corollary 13.5.4] (see [20]for a more refined version of this result).We end this article by giving two applications of Corollary 6.7. The next result andvarious related statements were recently shown in [16] via different methods. Corollary 6.10. Let w and w be two polynomially bounded weight functions on R n and let ≤ p , p < ∞ . Then, ( i ) M [ L p w b ⊗ π F L p w ] = M [ L w w ( F L p )] = W ( L p , L w w ) if p − + p − ≥ , ( ii ) M [ L p w b ⊗ ǫ F L p w ] = M [ C ,w w ( F L p )] = W ( L p , C ,w w ) if p − + p − ≤ ,topologically.Proof. In view of Corollary 6.7 (and Remark 6.9), the topological identities M [ L p w b ⊗ π F L p w ] = M [ L w w ( F L p )] , p − + p − ≥ , M [ L p w b ⊗ ǫ F L p w ] = M [ C ,w w ( F L p )] , p − + p − ≤ , follow from Proposition 5.26 and Proposition 5.30. The proof of the other two identitiesis straightforward and we omit them. (cid:3) Corollary 6.7 (and Remark 6.9) also imply that modulation spaces defined via TMIBsatisfy the sequential approximation property [23, Chapter 43]; we refer to [8] formore information on approximation properties for the classical modulation spaces M p,qw ,1 ≤ p, q < ∞ . Corollary 6.11. Let F be a TMIB on R n . 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Prangoski, Faculty of Mechanical Engineering, University Ss. Cyril and Method-ius, Karpos II bb, 1000 Skopje, Macedonia Email address :: c ∈ c ,w w (Λ ; ( F L p ) d (Λ )), there is a finite subset Λ (0)1 ofΛ such that, for all λ ∈ Λ \ Λ (0)1 , w ( λ ) w ( λ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ ∈ Λ ˜ c λ ,λ T λ ˜ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F L p ≤ (2 k χ k L p A A A A ) − ε =: ε . For any Λ (0)1 ⊆ Λ ′ , Λ ′′ ⊂ Λ , Λ ′ and Λ ′′ finite, and f ∈ K , f ∈ K , we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* f ⊗ f , X λ ∈ Λ ′ T λ χ ⊗ g λ − X λ ∈ Λ ′′ T λ χ ⊗ g λ +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X λ ∈ Λ ′ \ Λ (0)1 |h f , T λ χ i||h f , g λ i| + X λ ∈ Λ ′′ \ Λ (0)1 |h f , T λ χ i||h f , g λ i| . ABOR FRAME CHARACTERISATIONS OF MODULATION SPACES 33