The Homogeneous Approximation Property and localized Gabor frames

Transcription

1 Monatsh. Math. manuscript No. (will be inserted by the editor) The Homogeneous Approximation Property and localized Gabor frames Hans G. Feichtinger Markus Neuhauser Received: 17 April 2015 / Accepted: 9 June 2016 Abstract The Homogeneous Approximation Property (HAP) has been introduced in order to describe the locality of Gabor expansions in the Hilbert space L 2 (R d ). In this manuscript the HAP property is established for families of modulation spaces. Instead of the more recent theory of localized frames ([11]) which relies on Wiener pairs of Banach algebras of matrices, our approach is based on the constructive principles established in [5, 6, 10], using the fact that generalized modulation spaces are coorbit spaces with respect to the Schrödinger representation of the Heisenberg group (cf. [8]). For the (noncanonical) dual frames obtained constructively in this way the HAP property is verified. Keywords Homogeneous approximation property Gabor frames coorbit spaces modulation spaces Mathematics Subject Classification (2000) 42C15 42C40 46A16 46E10 46E30 1 Introduction We study the Homogeneous Approximation Property (HAP) for families of generalized modulation spaces. Any function in such a space which can be Corresponding author H. G. Feichtinger Faculty of Mathematics, Univ. Vienna, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria hans.feichtinger@univie.ac.at M. Neuhauser Lehrstuhl A für Mathematik, RWTH Aachen, D Aachen, Germany Department of Mathematics and Science, Faculty of Science, German University of Technology in Oman, P.O. Box 1816, Athaibah, PC 130, Sultanate of Oman markus.neuhauser@matha.rwth-aachen.de

2 2 H. G. Feichtinger and M. Neuhauser approximated by test functions has something like an essential support with respect to any given, potentially irregular, Gabor frame. HAP provides information about the size of this set in comparison with the approximation quality. Our results describe the invariance of this situation with respect to arbitrary shifts of the underlying point set which generates the irregular Gabor family. The investigation of the HAP for L 2 (R) was initiated by Ramanathan and Steger in [18], see also [16]. A survey of the history and evolution of the HAP, providing also connections to the Comparison Theorem, is found in [14]. The paper is organized as follows. Section 2 collects fundamental notions and concepts. In Section 3 we explain the HAP for families of spaces and give equivalent characterizations. Using Banach frames for modulation spaces as presented in [10, Theorem S], we show that the HAP is valid for families of generalized modulation spaces. In the last section, an irregular sampling theorem for a class of modulation spaces is presented. 2 Preliminaries and Notation As usual the translation operator is given by T x f(t) = f(t x), whereas frequency shifts or modulation operators, are given by M ω f(t) = e 2πiω t f(t) where ω t is the scalar product of ω R d and t R d. Both operators are unitary on L 2 (R d ). The time-frequency plane or phase space, resp., is the set R d R d and for λ = (x, ω) R d R d, time-frequency shifts are defined by π(λ) = M ω T x. The Short-Time Fourier Transform (STFT) of a function f L 2 (R d ) with respect to a window g L 2 (R d ) is given by V g f(x, ω) = f, M ω T x g L 2 (R d ) = f(t)e 2πiω t g(t x) dt. R d For Schwartz functions g S (R d ), the STFT extends naturally to the space S (R d ) of tempered distributions, with the bracket, understood as a dual sesqui-linear pairing between elements from S (R d ) and S (R d ) (see also [12]). We assume throughout this paper that (Y, Y ) is a solid and translation invariant Banach space of functions on phase space (compare [7]): Writing T λ for the translation operator on R 2d we request: (i) (Y, Y ) is continuously embedded into L 1 loc (Rd R d ); (ii) if G is a measurable function on R d R d, and G(λ) H(λ) almost everywhere for some H Y, then G Y and G Y H Y ; (iii) T λ H Y w(λ) H Y for all λ R d R d ; H Y where w is some fixed symmetric submultiplicative polynomially increasing weight function on R d R d such that F G Y F Y G 1,w for F Y, G L 1 w; and ( (iv) the space C c R d R ) d is dense in Y. The growth of the weight function w will dictate the necessary quality of the Gabor atom g allowed in the Gabor reconstruction theorem. For technical convenience we restrict our attention here to weight functions of polynomial

3 The Homogeneous Approximation Property and localized Gabor frames 3 growth, i. e. we assume that w(λ) = (1 + λ 2 ) s/2 for some s 0. For more general choices one has to use windows g from the spaces of test functions used to define ultra-distributions in order to obtain the main results of this paper in a slightly modified version. Our main example is a so-called weighted mixed-norm space. A submultiplicative weight w satisfies w(λ + λ ) w(λ) w(λ ), λ, λ R d R d. Furthermore a positive function m on R d R d is called w-moderate if m(λ + λ ) m(λ) w(λ ), λ, λ R d R d. For 1 p, q < the weighted mixed-norm space L p,q m (R 2d ) consists of those measurable functions F on R d R d for which the following norm is finite: ( ( ) ) q/p 1/q F L p,q := F (x, ω) p m(x, ω) p dx dω. m R d R d They are all solid and translation invariant Banach spaces on phase space. For generalized modulation spaces we fix a non-zero window function g S (R d ) and Y as above. The modulation space is defined as the space of all f S (R d ) with V g f Y, equipped with the norm f = V g f Y. It is a Banach space whose definition is independent of the choice of g. Choosing Y = L p,q m yields the classical modulation spaces denoted by Mm p,q (R d ). Other examples come from spaces of functions of variable bandwidth (see [1, 2]). Compare also [12, Chapter 11] and [3], for more details, or [4] for a nice historical survey. The density of the set Λ = (λ) of sampling points is defined to be the infimum over all δ > 0 such that R d R d = B δ(λ), where B δ (λ) stands for the open ball of radius δ centered at λ. A set Λ = (λ) is called separated if, for some δ > 0, λ λ implies B δ (λ) B δ (λ ) =. Finite unions of separated sets are called relatively separated. Given Y and Λ there is a corresponding (solid) Banach sequence space Y d := Y d (Λ) := {(c λ ) : c λ χ Bδ (λ) Y}, endowed with the norm (c λ ) Yd := c λ χ Bδ (λ) Y. Here ( χ Bδ (λ) denotes the characteristic function of the set B δ (λ). For Y = L ) p,q m R 2d and the lattice Λ = az d bz d these sequence spaces are simply the corresponding weighted natural mixed-norm l p -l q -sequence spaces. Coorbit theory (as developed in [5, 6, 10]), applied to (irregular) Gabor families, i. e. to families of the form (g λ ) := (π(λ)g) implies that they form Banach frames for any given g S (R d ) \ {0}, as long as the density of Λ is high enough, for families B w of generalized modulation spaces. This means that the coefficient mapping C : f ( f, g λ ) is uniformly bounded from Y to Y d on these spaces and has a left inverse of the form R : (c λ ) c λ g λ, see [10].

4 4 H. G. Feichtinger and M. Neuhauser On the other hand, sufficient separation of the point set Λ implies that one has Riesz projection bases for the same family of spaces. Here the roles are reversed. The synthesis mapping using (g λ ) has a stable left inverse, which provides the coefficients (cf. [9]). In the context of Banach frames the corresponding families ( g λ ) providing exact recovery are called dual frames whereas one can talk of biorthogonal families in the case of Riesz projection bases. The canonical choice is obtained by applying the inverse frame operator to the frame elements, or by obtaining the biorthogonal matrix via the inverse Gram matrix. While coorbit theory reveals that these mappings are universal for those families of spaces and the common constants depend only on the density of the point sets Λ and the quality of the Gabor atom g, the classical theory does not provide information about concentration. In fact, the norms of the (original and dual) frame elements in the weighted modulation spaces are not uniformly bounded. By discussing the translation invariance of the situation in this context we reveal the uniform concentration of dual frames obtained in this way, expressed in the form of the HAP. Finally let us note that the technical details relating the properties of the projective representation of phase space via time-frequency shifts to the properties of abstract coorbit space theory, in particular to the Schrödinger representation of the Heisenberg group, are provided in full detail in [8]. 3 Homogeneous Approximation Property (HAP) for Families In this section we introduce the concept of a Homogeneous Approximation Property for families of spaces. For the main result we need frames with certain properties. Given a (fixed) symmetric submultiplicative weight w on phase space we consider the family of all modulation spaces (understood in the wide sense, i. e. of the form) M (Y) such that Y satisfies the properties (i) (iv) from Section 2. We will denote this family by the symbol B w. It is one of the main messages of this note that HAP relevant statements can always be formulated uniformly for the whole family B w. For applications the following special case is of interest: Given a weight of the form w 0 (λ) := (1 + λ ) s0 for some s 0 0, the corresponding family B w0 contains all the (classical) modulation spaces Ms p,q (R d ), 1 p, q <, with s [ s 0, s 0 ], but also the Shubin classes Q s (R d ), for the same range of parameters. Even for s 0 = 0 one has uniform statements for the spaces M p,q 0, which are isometrically time-frequency invariant. Definition 1 Given a symmetric submultiplicative weight w on phase space, we consider pairs of dual frames for the family B w, i. e. a family of pairs

5 The Homogeneous Approximation Property and localized Gabor frames 5 {(g λ, g λ ) : λ Λ} forming an atomic decomposition of the form 1 f = f, g λ g λ, f S (R d ). We say that such a pair has the Homogeneous Approximation Property (HAP) with respect to B w, if for every ε > 0, B w, and f, there exists some compact set Q 0 R d R d such that any open Q Q 0 implies the following estimate, uniformly with respect to µ R d R d : f π(µ)f, g λ π(µ) 1 g λ < ε (1) (Q+µ) Remark 1 If π(µ) acts isometrically on M (Y) we are back to the original definition of the HAP. In other words, the size of the set of relevant sampling points for a reconstruction of π(µ)f (around its center, which moves with µ R d R d ) is of fixed size for the given level of precision. We distinguish between the following views on the HAP: The HAP, which considers the case L 2 (R d ) only (cf. [15,19]). The HAP for coorbit spaces which are isometrically π(µ)-invariant, such as the modulation spaces M p,q 0 (R d ). Statements concerning the HAP, for a pair of canonically dual frames (cf. [15,19]). Statements describing the quality of the frame in terms of localization of the frame or low correlation between the elements of the frame elements which are at some distance in phase space, resp. Statements which use strict coherence on the side of the analyzing frame, i. e. g λ = π(λ)g, or those allowing to switch the roles of g λ and g λ. Statements valid for ( g λ ) obtained by particular reconstruction methods. Further note that since π(µ) 1 = τπ( µ) for some τ C with τ = 1 we can rewrite condition (1), upon substituting µ = µ, as f f, π(µ )g λ π(µ ) g λ < ε. (Q µ ) Next, we discuss equivalent characterizations of the HAP for families. The following theorem describes a number of equivalent properties. Theorem 1 Given a symmetric submultiplicative weight w the following conditions for a pair of dual frames {(g λ, g λ ) : λ Λ} are equivalent: (i) HAP is valid, with respect to the family B w. 1 The atoms g λ are centered at λ (e. g. g λ = π(λ)g for some nice g); the family ( g λ ) need not be the canonical dual frame in our discussion.

6 6 H. G. Feichtinger and M. Neuhauser (ii) HAP is valid for every finite set {f 1,..., f N } in B w. (iii) HAP is valid for any compact subset K in B w. (iv) For every compact operator A on B w one has: For every ε > 0 there exists some compact set Q 0 R d R d, such that Q Q 0 and Q open implies Af π(µ)af, g λ π(µ) 1 g λ ε f, f. (Q+µ) Remark 2 It is easy to show that (i) above is equivalent to the choice of a rank one projection operator Af = f, h h, for some h L 2 ( R d) M(Y ) with h 2 = 1. Proof (i) (ii). Given ε > 0 and f 1,..., f N, there exist sets Q 1,..., Q N R d R d such that the HAP is satisfied for f j by Q j. Clearly then Q 0 = N j=1 Q j R d R d implies HAP for f 1,..., f N, simultaneously. (ii) (iii). Let K be a compact set in. For any f K we put B ε/3 (f) := {h K : f h < ε/3}. By compactness, there are f 1,... f N K with K N j=1 B ε/3(f j ). Hence, for f K there exists some 1 j N such that f f j < ε/3. Let Q 0 be such that the HAP for families holds for ε/3 and f 1,..., f N. Then for open Q Q 0 and µ := µ one has: f π(µ) 1 π(µ)f, g λ g λ (Q+µ) f f j + f j f j, π(µ )g λ π(µ ) g λ + (Q µ ) + f j f, π(µ )g λ π(µ ) g λ < ε. (Q µ ) The first and second term in the last sum are less than ε/3 by assumption. In order to estimate the last term in the preceding estimate we recall that the families {(π(µ )g λ, π(µ ) g λ ) : λ Λ} form a pair of dual atoms. Therefore this third term can be estimated by f j f, π(µ )g λ π(µ ) g λ < ε/3, noting that Y d (Λ), containing the range of the operator C, is solid. (iii) (iv). Let B = {f : f = 1}. By the compactness of the operator K := A(B) is relatively compact. Given ε > 0, we know by (iii) that there is a compact set Q 0 such that one has for all f B and all open Q Q 0 : Af π(µ)af, g λ π(µ) 1 g λ ε. (Q+µ)

7 The Homogeneous Approximation Property and localized Gabor frames 7 For arbitrary f \ {0} we obtain the claim as f 1 f B and the claim follows by multiplication with f. For f = 0 it was obvious from the beginning. (iv) (i). We may suppose f 0 and define A(h) = f 1 h, f f. This is a rank 1 and hence a compact operator. By (iv) we know that for ε/2 there is a Q 0 such that the equation in (iv) holds for all open Q Q 0 and specializing to h = f 1 f this means that: f π(µ)f, g λ π(µ) 1 g λ ε 2 < ε (Q+µ) Next, we show that the HAP for families is valid for the family of modulation spaces B w as described in the original papers on coorbit theory, and for the reconstruction principles described via iterative procedures in [5, 6]. Although no concrete dual families occur in the original papers they are implicitly around, and depend to some extent on the used ingredients (e. g. some BUPUs i. e. bounded uniform partitions of unity) in the process, but the statements made below are valid for all of the situations to which coorbit reconstruction methods apply. Whereas methods from localization theory make a statement which applies to a larger class of cases, it is restricted to the specific choice of the canonical dual frame. In both cases more quantitative variants could be of interest. Let us start by recalling a Banach frame result due to Gröchenig [10, Theorem S]. We state it here only for generalized modulation spaces and without exploiting all technical assumptions. For δ > 0 we use the local maximal function F (λ) := sup λ B δ (λ) F (λ ) and the δ-oscillation of a function F given by osc δ F (λ) := sup F (λ ) F (λ). λ B δ (λ) Proposition 1 Let w be a ( symmetric submultiplicative weight on the TFplane. Then for any g M ) w 1 R d there is some δ > 0 such that osc δ (V g g) 1,w < 1/ V g g 1,w. Given a family B w of modulation spaces such that for all B w the spaces Y satisfy the conditions (i) (iv) from Section 2. Then for any such atom g and any δ-dense and relatively separated set Λ there exists some dual family ( g λ ) allowing stable reconstruction (the sum being convergent in ): f = V g f(λ) g λ for all f B w. Remark 3 The (non-canonical) dual frame ( g λ ) depends on some bounded uniform partition of unity Ψ. But each of these families has the HAP for B w.

8 8 H. G. Feichtinger and M. Neuhauser We rephrase the previous result on Banach frames in order to obtain uniformity with respect to location. In fact, we can make use (once more) of the group structure behind coorbit theory. By virtue of the homomorphism property of π we obtain f = π(µ) 1 π(µ)f = V g [π(µ)f] (λ)π(µ) 1 g λ, where we use the expansion for π(µ)f instead of f. Remark 4 A different way to define a HAP for families would be to require π (µ) f π (µ) f, g λ g λ < ε (2) (Q+µ) instead of (1). Note that in the case of isometrically π-invariant coorbit spaces (one could say this is the class B w0, with w 0 (λ) 1), among them are the modulation spaces M p,q 0 (R d ), this is equivalent to our definition of HAP for families. For the weighted case this variant would be less meaningful. We are now in a position to formulate the main result of the paper. It states that the HAP for families is valid in the context of generalized modulation spaces. Note that ( g λ ) will in general not be the canonical dual frame. Theorem 2 In the situation described in Proposition 1, with the dual frame ( g λ ) arising from coorbit theory, the HAP is valid with respect to the family B w : For any B w the following is true: Given f and ε > 0 there is a compact set Q 0 R d R d such that Q Q 0 and Q open implies: f [ V g π(µ) 1 f ] (λ) π(µ) g λ < ε, (Q µ) uniformly with respect to µ R d R d. Equivalently, one can claim that (up to an ε error) a good reconstruction of f can be achieved, using only those samples of the STFT of f in Λ + µ which are located within Q, because < ε, f λ+µ Q V g f(λ + µ) e 2πiλ2 x π(µ) g λ with λ = (λ 1, λ 2 ) and µ = (x, ω), λ 1, x R d, λ 2, ω R d. Proof We use a uniform partition of unity Ψ = (ψ λ ), of size δ, satisfying (i) ψ λ is continuous, with supp(ψ λ ) B δ (λ), (ii) 0 ψ λ (λ ) 1 for all λ R d R d, (iii) ψ λ(λ ) = 1 for all λ R d R d.

9 The Homogeneous Approximation Property and localized Gabor frames 9 For given µ we consider the shifted versions of Ψ, i. e. Ψ µ = (T µ (ψ λ )). This is again a partition of unity of size δ with centers λ+µ as supp(t µ (ψ λ )) B δ (λ + µ) and our new set Λ is Λ + µ. Further we define approximation operators by Σ µ : W (L, Y) Y and S µ : W (L, Y) Y V g g Σ µ (F ) = F (λ + µ)t µ (ψ λ ) ( ) S µ (F ) = Σ µ (F ) V g g = F (λ + µ) T µ (ψ λ ) V g g = = F (λ + µ)t µ (ψ λ ) V g g. Hence Σ µ = T µ Σ 0 T µ and accordingly S µ = T µ S 0 T µ. We have that Σ µ (F ) (λ ) = F (λ + µ)t µ (ψ λ ) (λ ) F (λ + µ) T µ (ψ λ ) (λ ) F (λ )T µ (ψ λ ) (λ ) = F (λ ) as T µ (ψ λ ) (λ ) 0 only if λ µ λ < δ. Since F W (L, Y) we have F Y. By solidity of Y we have Σ µ (F ) Y. Hence Σ µ is well-defined. Thus S µ (F ) Y = Σ µ (F ) V g g Y Σ µ (F ) Y V g g 1,w F Y V g g 1,w. In [10] the approximation operator S µ is denoted by S Ψµ. The conditions in Proposition 1 imply the invertibility of S µ on Y V g g. And we have: F (λ ) F (λ + µ)t µ ψ λ (λ ) sup F (λ ) F (ν) T µ ψ λ (λ ) ν B δ (λ ) = osc δ (F )(λ )T µ ψ λ (λ ) = osc δ (F )(λ ). Let J : Y Y V g g be convolution with V g g on the right. The assumptions on V g g ensure that J is uniformly bounded on the family B w. If F Y V g g then using the preceding estimates we obtain: JF S µ F Y F V g g F (λ + µ)t µ ψ λ V g g Y ( = F ) F (λ + µ)t µ ψ λ V g g osc δ (F ) Y V g g 1,w F osc δ (V g g) Y V g g 1,w F Y osc δ (V g g) 1,w V g g 1,w, Y

10 10 H. G. Feichtinger and M. Neuhauser where H (λ) = H( λ). Denote by Y g the space V g (M (Y)) = Y V g g. The convolution J is the identity on Y g and T µ J T µ = J. The function g is chosen such that osc δ (V g g) 1,w V g g 1,w < 1. Consequently for any µ: (J S µ ) Yg osc δ (V g g) 1,w V g g 1,w = κ < 1. Using the Neumann representation R µ = k=0 (J S µ) k for the inverse on Y g one has R µ = T µ R 0 T µ and the norm estimate R µ ( 1 (J S µ ) Yg ) 1 (1 κ) 1 <. We denote by Vg 1 the inverse operator from Y g onto M (Y) and define g λ = Vg 1 (R 0 (ψ λ V g g)). For ε > 0 we find by property (iv) from Section 2 ( an H C c R d R ) d such that: V g f (1 κ) ε H <. Y V g g 1,w Let Q 1 R d R d be the compact support of H. Then (1 χ Q1 ) V g f ( ) Y = (1 χ Q1 ) V g f Y H V g f (1 κ) ε H < Y V g g 1,w using the solidity of Y. This ensures that in the end the approximation will be good enough. Due to the fact that V g is an intertwining operator between π and T and that T µ R 0 = R µ T µ we have π (µ) g λ = π (µ) ( V 1 g (R 0 (ψ λ V g g)) ) = Vg 1 (T µ (R 0 (ψ λ V g g))) = Vg 1 (R µ (T µ (ψ λ V g g))). { } Then Q 0 = Q 1 + B δ (0) = λ + µ : λ Q 1, µ B δ (0) is still compact and for open Q Q 0 we have f V g f(λ + µ) e 2πiλ2 x π(µ) g λ = λ+µ Q [ = f V g π(µ) 1 f ] (λ)π(µ) g λ = V g f (Q µ) (Q µ) V g [π (µ) 1 f] (λ) R µ (T µ (ψ λ ) V g g) Y.

11 The Homogeneous Approximation Property and localized Gabor frames 11 Since V g f W (L, Y) we also have χ Q V g f W (L, Y) by solidity of W (L, Y). Hence we can apply S µ and obtain V g [π (µ) 1 f] (λ) R µ (T µ (ψ λ ) V g g) = (R µ S µ ) (χ Q V g f). (Q µ) Thus the expression we wanted to estimate now becomes f V g f(λ + µ) e 2πiλ2 x π(µ) g λ = λ+µ Q = V g f (R µ S µ ) (χ Q V g f) Y = R µ (S µ (V g f χ Q V g f)) Y R µ S µ (V g f χ Q V g f) Y (1 κ) 1 ((1 χq ) V g f ) Y V g g 1,w (1 κ) 1 (1 χq1 ) V g f Y V g g 1,w < ε by solidity of Y since ((1 χ Q ) V g f (λ) ) (1 χ Q1 ) V g f (λ) for any open Q Q 0. Remark 5 A close inspection of the proof shows that if a given function f is contained in a parameterized family of spaces M (Y) B w then Q 0 can be chosen uniformly with respect to all the norms of these spaces, at least if f is normalized properly. 4 A Localized Piano Reconstruction Theorem Finally we look at certain irregular sampling theorems in the Gabor context. First let us recall the so-called Piano Reconstruction Theorem (compare [8, Theorem 25, Part V]), stated in the terminology of Riesz projection bases. We cite Theorem 25, Part V in [8], but for simplicity we describe the unweighted case, using our notation. We also write h γ for π(γ)h in the sequel. Theorem 3 Given h S 0 (R d ) there exists S > 0 such that the following holds true: If a sequence of points (γ) γ Γ in R 2d is 2S-separated, i. e. B S (γ) B S (γ ) = for γ γ, then it is possible to recover the unique coefficients of any f of the form f = γ Γ c γ h γ = γ Γ c γ π(γ)h, with sup c γ < (3) γ Γ in a stable way, using the biorthogonal family ( h γ ) γ Γ in S 0 (R d ), via c γ = C(f) := f, h γ.

12 12 H. G. Feichtinger and M. Neuhauser In fact, the linear mapping C : f C(f) is well defined on all of S 0 (R d ), and maps the Banach triple (S 0, L 2, S 0 )(R d ) into the corresponding triple (l 1, l 2, l )(Γ ) in a bounded way and f γ Γ c γh γ is just the projection onto the closed linear span of the Gabor Riesz basic sequence (h γ ) γ Γ. Whereas the HAP implies that individual elements h γ can be approximately recovered from the samples of the STFT of V g (h)(λ), using points λ around γ, we claim now that the recovery of f is possible even for (potentially infinite) linear combinations of such elements, as described in Theorem 3 (but now h, Γ are assumed to be given and known to the user). Obviously the overlapping supports of the contributions V g (h γ ) will be of relevance. In order to avoid too many technical details we restrict our argument to the S 0 (R d )-setting, i. e. we want to show how to recover f V 1, the closed linear span of (h γ ) γ Γ within ( S 0 (R d ), S0 ), from samples around Γ. Theorem 4 Given 0 h S 0 (R d ) and Γ as above, sufficiently well separated. ( We consider ) the closed linear span V 1 of the family (π(γ)h) γ Γ in S0 (R d ), S0, which coincides with the l 1 -span of this family. Then there exists T > 0 such that every f V 1 can be completely recovered from the samples of V g (f)(λ) at λ Λ (Γ + B T (0)) only. The proof relies on the fact that the HAP principle allows us to recover the individual contributions locally and glue the information together. Then the fact that we have a projection onto the closed subspace V 1 S 0 (R d ) can be used to start an iterative algorithm which leads to complete recovery. ( We will need a specific version of the estimate (2). Choosing there Y = L 1 R d R ), d f = h and the open set Q (depending on Γ now, and not just on the individual point γ Γ ) as Q = Γ + B T (0), where T is chosen such that Q 0 as arising in Theorem 2 satisfies Q 0 B T (0). Proposition 2 Given h S 0 (R d ) and ε > 0 as well as a Gabor frame (g λ ) as in Theorem 3 above, coorbit theory provides a bounded family ( g λ ) in ( ) S 0 (R d ), S0 some radius T > 0 such that for any subset Λ Λ, with Λ (Λ (B T (γ))) one has π(γ)h V g (π(γ)h)(λ) g λ < ε. S 0 In order to simplify the notations we set Λ T (γ) := Λ B T (γ), T > 0, for the segment of Λ near γ. Correspondingly we write Λ T (Γ ) for the union of all these subsets (over the index set Γ ). Proof (of Theorem 4) First we describe the approximate recovery. Writing f T for the local reconstruction f T := T (Γ ) f, g λ g λ we have f f T = \Λ T (Γ ) V g (f)(λ) g λ.

13 The Homogeneous Approximation Property and localized Gabor frames 13 Combined with (3) we have to estimate f c γ h γ, g λ g λ = γ Γ γ Γ T (Γ ) c γ \Λ T (Γ ) h γ, g λ g λ For each fixed γ Γ the index set Λ \ Λ T (Γ ) is a subset of the complement of Λ T (γ) and thus we can estimate the S 0 -norm of the difference by ε if T > 0 is chosen appropriately, according to Proposition 2. Altogether we obtain the following estimate for the approximation error f f T S0 c γ ε (C ε ) f S0, γ Γ where C is the equivalence constant between the l 1 (Γ )-norm of the coefficient sequence (c γ ) γ Γ and the S 0 -norm of f. But f T and hence the difference f f T belong to the same Banach space V 1 (the closed linear span of (π(γ)h) γ Γ within ( S 0 (R d ), S0 ) ). And therefore the mapping BT : f f T is invertible on V 1, if ρ = C ε < 1. In closed form we just have f = B 1 T (B T (f)) = T (Γ ) V g (f)(λ) B 1 T ( g λ), where obviously the sum is absolutely convergent in ( S 0 (R d ), S0 ). Remark 6 Of course, if the situation described in Theorem 4 applies (for fixed h, Γ and T ), then it also applies to any subset of Γ. Let us therefore assume that Γ 1 and Γ 2 are two disjoint subsets of Γ. That two users making use of a mobile channel T (close to the identity operator) may use the two Gaborian Riesz basic sequences (h γ ) γ Γ1 and (h γ ) γ Γ2, thus sending (simultaneously) two signals f 1 and f 2 (both of the form (3)). Then the corresponding receivers have to sample the spectrogram V g (f) of the received signal T(f) f, with f = f 1 + f 2 only near Γ 1 or Γ 2 respectively, i. e. they can recover the coefficient sequences (c γ ) γ Γ1 and (c γ ) γ Γ2 perfectly. Of course in practice the mobile channel will distort the signals h γ a little bit, because they are only approximate eigenvectors to a slowly varying channel T (describing the distortion between sending station and receiver, see [17] or [20]). Acknowledgements The authors were supported by the Marie-Curie Excellence Grant MEXT-CT (EUCETIFA). The authors would like to thank the reviewers for their comments which have helped to essentially improve the presentation of our results. References 1. R. Aceska. Functions of variable bandwidth: A time-frequency approach. Ph.D. thesis, Vienna, R. Aceska, H. G. Feichtinger. Functions of variable bandwidth via time-frequency analysis tools. J. Math. Anal. Appl. 382(1), (2011).

14 14 H. G. Feichtinger and M. Neuhauser 3. H. G. Feichtinger. Modulation spaces of locally compact Abelian groups. In R. Radha, M. Krishna, and S. Thangavelu, editors, Proc. Internat. Conf. on Wavelets and Applications, January 2002, pages New Delhi Allied Publishers, Chennai, H. G. Feichtinger. Modulation Spaces: Looking back and ahead. Sampl. Theory Signal Image Process. 5(2), (2006). 5. H. G. Feichtinger, K. Gröchenig. Banach spaces related to integrable group representations and their atomic decompositions I. J. Funct. Anal. 86, (1989). 6. H. G. Feichtinger, K. Gröchenig. Banach spaces related to integrable group representations and their atomic decompositions II. Monatsh. Math. 108, (1989). 7. H. G. Feichtinger, K. Gröchenig. Irregular sampling theorems and series expansions of band-limited functions. J. Math. Anal. Appl. 167(2), (1992). 8. H. G. Feichtinger, K. Gröchenig. Gabor wavelets and the Heisenberg group: Gabor expansions and Short Time Fourier transform from the group theoretical point of view. In C. Chui, editor, Wavelets A Tutorial in Theory and Applications, volume 2 of Wavelet Anal. Appl., pages Academic Press, Boston, H. G. Feichtinger, G. Zimmermann. A Banach space of test functions for Gabor analysis. In H. Feichtinger and T. Strohmer, editors, Gabor analysis and algorithms: Theory and applications, Applied and Numerical Harmonic Analysis, pages Birkhäuser, Boston, K. Gröchenig. Describing functions: Atomic decompositions versus frames. Monatsh. Math. 112, 1 41 (1991). 11. K. Gröchenig. Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl. 10(2), (2004). 12. K. Gröchenig. Foundations of Time-Frequency Analysis. Appl. Numer. Harmon. Anal. Birkhäuser, Boston, K. Gröchenig. The Homogeneous Approximation Property and the Comparison Theorem for coherent frames. Sampl. Theory Signal Image Process. 7(3), (2008). 14. C. Heil. The Density Theorem and the Homogeneous Approximation Property for Gabor frames. In P. E. T. Jorgensen, K. D. Merrill, and J. D. Packer, editors, Representations, Wavelets, and Frames, pages Birkhäuser, Boston, C. Heil, G. Kutyniok. The Homogeneous Approximation Property for wavelet frames and Schauder bases. J. Approx. Theory 147(1), (2007). 16. J. A. Hogan, J. D. Lakey. Time-Frequency and Time-Scale Methods. Adaptive Decompositions, Uncertainty Principles, and Sampling. Birkhäuser, Boston, T. Hrycak, S. Das, G. Matz, and H. G. Feichtinger. Practical estimation of rapidly varying channels for OFDM systems. IEEE Trans. Comm. 59(11), J. Ramanathan, T. Steger. Incompleteness of sparse coherent states. Appl. Comput. Harmon. Anal. 2(2), (1995). 19. W. Sun, B. Liu. Homogeneous Approximation Property for continuous wavelet transforms. J. Approx. Theory 155(2), (2007). 20. G. Tauböck, M. Hampejs, G. Matz, F. Hlawatsch, and K. Gröchenig. LSQR-based ICI equalization for multicarrier communications in strongly dispersive and highly mobile environments. In Proc. IEEE SPAWC 07: Signal Processing Advances in Wireless Communications, 8th Workshop on, pages 1 5, Helsinki, 2007.