Gabor frames with atoms in $$M^q(\mathbb{R})$$ but not in $$M^p(\mathbb{R})$$ for any $$1\leq p < q\leq 2$$

A Gabor or Weyl-Heisenberg frame for L 2 (R d ) is generated by time-frequency shifts of a square-integrable function, the Gabor atom, along a time-frequency lattice. The dual frame is again a Gabor frame, generated by the dual atom. In general, Gabor frames are not stable under a perturbation of the lattice constants; that is, even for arbitrarily small changes of the parameters the frame property can be lost. In contrast, as a main result we show that this kind of stability does hold for Gabor frames generated by a Gabor atom from the modulation space M 1 (R d ), which is a dense subspace of L 2 (R d ). Moreover, in this case the dual atom depends continuously on the lattice constants. In fact, we prove these results for more general weighted modulation spaces. As a consequence, we obtain for Gabor atoms from the Schwartz class that the continuous dependence of the dual atom holds even in the Schwartz topology. Also, we complement these main results by corresponding statements for Gabor Riesz sequences and their biorthogonal system.

Let ψ \psi be a frame vector under the action of a collection of unitary operators U \mathcal U . Motivated by the recent work of Frank, Paulsen and Tiballi and some application aspects of Gabor and wavelet frames, we consider the existence and uniqueness of the best approximation by normalized tight frame vectors. We prove that for any frame induced by a projective unitary representation for a countable discrete group, the best normalized tight frame (NTF) approximation exists and is unique. Therefore it applies to Gabor frames (including Gabor frames for subspaces) and frames induced by translation groups. Similar results hold for semi-orthogonal wavelet frames.

In this paper, we study multivariate Gabor frames in matrix-valued signal spaces over locally compact abelian (LCA) groups, where the lower frame condition depends on a bounded linear operator [Formula: see text] on the underlying matrix-valued signal space. This type of Gabor frame is also known as a multivariate [Formula: see text]-Gabor frame. By extending work of G[Formula: see text]vruta, we present necessary and sufficient conditions for the existence of [Formula: see text]-Gabor frames of multivariate matrix-valued Gabor systems. Some operators which can transform multivariate matrix-valued Gabor and [Formula: see text]-Gabor frames into [Formula: see text]-Gabor frames in terms of adjointable operators are discussed. Finally, we give a Paley–Wiener-type perturbation result for multivariate matrix-valued [Formula: see text]-Gabor frames.

G?vruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely ?-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operator?. For a locally compact abelian groupGand a positive integer n, westudy frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space L2(G,Cn?n) , where a bounded linear operator ? on L2(G,Cn?n) controls not only lower but also the upper frame condition. We term such frames matrix-valued (?,?*)-Gabor frames. Firstly, we discuss frame preserving mapping in terms of hyponormal operators. Secondly, we give necessary and sufficient conditions for the existence of matrix-valued (?,?*)- Gabor frames in terms of hyponormal operators. It is shown that if ? is adjointable hyponormal operator, then L2(G,Cn?n) admits a ?-tight (?,?*)-Gabor frame for every positive real number ?. A characterization of matrix-valued (?,?*)-Gabor frames is given. Finally, we show that matrix-valued (?,?*)-Gabor frames are stable under small perturbation of window functions. Several examples are given to support our study.

We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions H n . Let h = (H 0, H 1, . . . , H n ) be the vector of the first n + 1 Hermite functions. We give a complete characterization of all lattices $${\Lambda \subseteq \mathbb{R} ^2}$$ such that the Gabor system $${\{ {\rm e}^{2\pi i \lambda _{2} t}{\bf h} (t-\lambda _1): \lambda = (\lambda _1, \lambda _2) \in \Lambda \}}$$ is a frame for $${L^2 (\mathbb{R} , \mathbb{C} ^{n+1})}$$ . As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor frame and a new estimate for the lower frame bound. The main tools are growth estimates for the Weierstrass σ-function, a new type of interpolation problem for entire functions on the Bargmann–Fock space, and structural results about vector-valued Gabor frames.

Let \(\Lambda=\mathcal{K}\times\mathcal{L}\) be a full rank time-frequency lattice in ℝd×ℝd. In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L2(ℝd) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)≤1, and to a dual Gabor Riesz basis pair for a Λ-shift invariant subspace containing M when v(Λ)>1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7:419–433, [2001]) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel–Gabor family G(g,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ)∪(∪j=1NG(gj,Λ)) for L2(ℝd). We show that this is true whenever v(Λ)≤N. In particular, when v(Λ)≤1, any Bessel–Gabor system is a subset of a tight Gabor frame G(g,Λ)∪G(h,Λ) for L2(ℝd). Related results for affine systems are also discussed.

Gabor expansions of discrete signals and images have a wide range of applications in signal analysis and pattern recognition. It is well known that the difficulty in expanding a 1-D or a 2-D signal into a Gabor series is the non-orthogonality of the building blocks, which are time-frequency shifted versions (along some lattice in the TF-plane) of a, given Gabor atom. The theory of frames (Gabor or Weyl-Heisenberg frames) has reached fame as the appropriate tool for resolving the problem. In particular, the dual frame turns out to be again a Gabor frame with respect to the same lattice, and its generating function is called the dual Gabor atom g/spl tilde/. It is obtained by applying the inverse of the frame operator S to the original frame (or just the Gabor atom), i.e. g/spl tilde/= S/sup -1/g. There is also another important use for the dual Gabor window. Given the sampled STFT (short time or sliding window Fourier transform) of some signal x with respect to the window g over the same lattice it is possible to recover the signal x using a simple Shannon-type reconstruction formula. We present some basic ideas behind a new family of iterative algorithms for determining the dual Gabor atom in the finite (discrete and periodic) case. They are mainly based on the conjugate gradient methods in combination with structural properties of the Gabor frame operator. >

The theory of Gabor frames has been extensively investigated. This paper addresses partial Gabor systems. We introduce the concepts of partial Gabor system, frame and dual frame. We present some conditions for a partial Gabor system to be a partial Gabor frame, and using these conditions, we characterize partial dual frames. We also give some examples. It is noteworthy that the density theorem does not hold for general partial Gabor systems.

Vector-valued Gabor frames associated with periodic subsets of the real line

We attempt to analyze Gabor frames for L2 spaces on finite non cyclic abelian groups from their natural perspectives and establish their equivalence with Gabor frames on the isomorphic product groups of finite cyclic groups by means of unitary and non unitary invertible linear transformations. Gabor frames and their canonical dual frames on finite non cyclic groups are identified as the images of the same Gabor frame on a finite product of finite cyclic groups under invertible linear transformations.

It is a well known fact that the time-frequency domain is very well adapted for representing audio signals. The main two features of time-frequency representations of many classes of audio signals are sparsity (signals are generally well approximated using a small number of coefficients) and persistence (significant coefficients are not isolated, and tend to form clusters). This contribution presents signal approximation algorithms that exploit these properties, in the framework of hierarchical probabilistic models. Given a time-frequency frame (i.e. a Gabor frame, or a union of several Gabor frames or time-frequency bases), coefficients are first gathered into groups. A group of coefficients is then modeled as a random vector, whose distribution is governed by a hidden state associated with the group. Algorithms for parameter inference and hidden state estimation from analysis coefficients are described. The role of the chosen dictionary, and more particularly its structure, is also investigated. The proposed approach bears some resemblance with variational approaches previously proposed by the authors (in particular the variational approach exploiting mixed norms based regularization terms). In the framework of audio signal applications, the time-frequency frame under consideration is a union of two MDCT bases or two Gabor frames, in order to generate estimates for tonal and transient layers. Groups corresponding to tonal (resp. transient) coefficients are constant frequency (resp. constant time) time-frequency coefficients of a frequency-selective (resp. time-selective) MDCT basis or Gabor frame.

We study time-continuous Gabor frame generating window functions g satisfying decay properties in time and/or frequency with particular emphasis on rational time-frequency lattices. Specifically, we show under what conditions these decay properties of g are inherited by its minimal dual γ0 and by generalized duals γ. We consider compactly supported, exponentially decaying, and faster than exponentially decaying (i.e., decay like |g(t)|≤Ce−α|t| 1/α for some 1/2≤α<1) window functions. Particularly, we find that g and γ0 have better than exponential decay in both domains if and only if the associated Zibulski-Zeevi matrix is unimodular, i.e., its determinant is a constant. In the case of integer oversampling, unimodularity of the Zibulski-Zeevi matrix is equivalent to tightness of the underlying Gabor frame. For arbitrary oversampling, we furthermore consider tight Gabor frames canonically associated to window functions g satisfying certain decay properties. Here, we show under what conditions and to what extent the canonically associated tight frame inherits decay properties of g. Our proofs rely on the Zak transform, on the Zibulski-Zeevi representation of the Gabor frame operator, on a result by Jaffard, on a functional calculus for Gabor frame operators, on results from the theory of entire functions, and on the theory of polynomial matrices.

We develop an alternative approach to the study of Fourier series, based on the Short-Time-Fourier Transform (STFT) acting on $$L_{\nu }^{2}(0,1)$$ L ν 2 ( 0 , 1 ) , the space of measurable functions f in $$\mathbb {R}$$ R , square-integrable in (0, 1), and time-periodic up to a phase factor: for fixed $$\nu \in \mathbb {R}$$ ν ∈ R , $$\begin{aligned} f(t+k)=e^{2\pi ik\nu }f(t){, \ }k\in \mathbb {Z}\text {.} \end{aligned}$$ f ( t + k ) = e 2 π i k ν f ( t ) , k ∈ Z . The resulting phase space is the vertical strip $$\mathbb {C}/\mathbb {Z}=[0,1)\times \mathbb {R}$$ C / Z = [ 0 , 1 ) × R , a flat model of an infinite cylinder, which leads to Gabor frames with an interesting structure theory, allowing for a Janssen-type representation. As expected, a Gaussian window leads to a Fock space of entire functions, studied in the companion paper by the same authors [Beurling-type density theorems for sampling and interpolation on the flat cylinder]. When g is a Hermite function, we are lead to true Fock spaces of polyanalytic functions (Landau level eigenspaces) on the vertical strip $$[0,1)\times \mathbb {R}$$ [ 0 , 1 ) × R . We first prove a density condition for a lattice to be interpolating in this space. Furthermore, an analogue of the sufficient Wexler-Raz conditions is obtained which leads to new criteria for Gabor frames in $$L^{2}(\mathbb {R})$$ L 2 ( R ) , and to sufficient conditions for Gabor frames in $$L_{\nu }^{2}(0,1)$$ L ν 2 ( 0 , 1 ) with Hermite windows (an analogue of a theorem of Gröchenig and Lyubarskii about Gabor frames with Hermite windows) and with totally positive windows in the Feichtinger algebra (an analogue of a recent theorem of Gröchenig). We also consider a vectorial STFT in $$L_{\nu }^{2}(0,1)$$ L ν 2 ( 0 , 1 ) and, using the vector with the first Hermite functions as window, we introduce the (full) Fock spaces of polyanalytic functions on $$[0,1)\times \mathbb {R}$$ [ 0 , 1 ) × R and their associated Bargmann-type transforms, and prove an analogue of Vasilevski’s orthogonal decomposition into true polyanalytic Fock spaces (Landau level eigenspaces on $$[0,1)\times \mathbb {R}$$ [ 0 , 1 ) × R ). We conclude the paper with an analogue of Gröchenig-Lyubarskii’s sufficient condition for Gabor super-frames with Hermite functions, which is equivalent to a sufficient sampling condition on the full Fock space of polyanalytic functions on $$[0,1)\times \mathbb {R}$$ [ 0 , 1 ) × R . The proofs of the results about Gabor frames, involving some of Gröchenig’s most significant results of the past 25 years, are a clear indication of his influence on the field during this period.

Recently, Gabor analysis on locally compact abelian (LCA) groups has interested some mathematicians. The half real line is an LCA group under multiplication and the usual topology. This paper addresses spline Gabor frames for , where is the corresponding Haar measure. We introduce the concept of spline functions on by ‐convolution and estimate their Gabor frame sets, that is, lattice sets such that spline generating Gabor systems are frames for . For an arbitrary spline Gabor frame with special lattices, we present its one dual Gabor frame window, which has the same smoothness as the initial window function. For a class of special spline Gabor Bessel sequences, we prove that they can be extended to a tight Gabor frame by adding a new window function, which has compact support and same smoothness as the initial windows. And we also demonstrate that two spline Gabor Bessel sequences can always be extended to a pair of dual Gabor frames with the adding window functions being compactly supported and having the same smoothness as the initial windows.

Gabor frames have been extensively studied in time-frequency analysis over the last 30 years. They are commonly used in science and engineering to synthesize signals from, or to decompose signals into, building blocks which are localized in time and frequency. This chapter contains a basic and self-contained introduction to Gabor frames on finite-dimensional complex vector spaces. In this setting, we give elementary proofs of the central results on Gabor frames in the greatest possible generality; that is, we consider Gabor frames corresponding to lattices in arbitrary finite Abelian groups. In the second half of this chapter, we review recent results on the geometry of Gabor systems in finite dimensions: the linear independence of subsets of its members, their mutual coherence, and the restricted isometry property for such systems. We apply these results to the recovery of sparse signals, and discuss open questions on the geometry of finite-dimensional Gabor systems.