Gabor Frames: Characterizations and Coarse Structure

Gröchenig, Karlheinz; Koppensteiner, Sarah

doi:10.1007/978-3-030-32353-0_4

Cited by 9 publications

(2 citation statements)

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“…We present a reduced form and refer to the results of Daubechies, Landau and Landau [10], Janssen [34], Ron and Shen [44] and Wexler and Raz [48]. For a more thorough treatise of the topic we refer to [26], where we also find the following statement:…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Gabor frame bound optimizations

Faulhuber¹,

Irina²

2022

Preprint

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Section: Notation and Preliminariesmentioning

confidence: 99%

Gabor frame bound optimizations

Faulhuber¹,

Irina²

2022

Preprint

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“…A large number of different frame constructions are used in harmonic analysis both in practical applications such as image denoising [9,45,5,53], restoring truncated signals [51,52], edge detection [50,41] and compressed sensing [1] and in pure mathematics for characterising function spaces in terms of the frame coefficients [44,64,27,20,58], studying the boundedness of operators [28,47,46,37] or for characterising the wave front set of distributions [17,39,8]. The most important of these frame constructions are wavelets [14], Gabor frames [27,29], shearlets [38,40], curvelets [7], ridgelets [6,30] and wave atoms [15].…”

Section: Introductionmentioning

confidence: 99%

Design and properties of wave packet smoothness spaces

Bytchenkoff

Voigtlaender

2020

Journal de Mathématiques Pures et Appliquées

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We introduce a family of quasi-Banach spaces -which we call wave packet smoothness spacesthat includes those function spaces which can be characterised by the sparsity of their expansions in Gabor frames, wave atoms, and many other frame constructions. We construct Banach frames for and atomic decompositions of the wave packet smoothness spaces and study their embeddings in each other and in a few more classical function spaces such as Besov and Sobolev spaces. RésuméNous introduisons une famille d'espaces affines quasi-normés complets -que nous appellerons espaces de paquets d'ondelettes réguliers -qui incluent de nombreux espaces de fonctions caractérisés par leurs transformées, comme celle de Gabor ou en ondelettes, clairsemées. Nous construisons des cadres de Banach et des décompositions atomiques pour ces espaces etétudions leurs inclusions l'un dans l'autre ainsi que dans quelques espaces de fonctions devenus classiques tels que ceux de Sobolev ou de Besov.(2.1)To ensure that the decomposition space D(Q, L p , q w ) is indeed a well-defined quasi-Banach space, certain conditions must be imposed on the covering Q, the partition of unity (ϕ i ) i∈I and the weight w = (w i ) i∈I . These conditions and elementary properties of the decomposition spaces D(Q, L p , q w ) will be reminded in Section 2.1.An attractive feature of decomposition spaces is the recently developed theory of structured Banach frame decompositions of decomposition spaces [63], which shows that there is a close connexion between the existence of a sparse expansion of a given function f in terms of a given frame, and the membership of f in a certain decomposition space, which depends on the frame under consideration.This theory will be formally introduced in Section 2.2; here, we outline the underlying intuition. We first note that most frame constructions used in harmonic analysis have two crucial properties:• The frame is a generalised shift-invariant system (see [34,59,11,21,43] for more about these systems), i.e., it is of the form Ψ = (L x ψ j ) j∈I,x∈Γ j for suitable generators (ψ j ) j∈I and certain lattices Γ j = δB j Z d where the matrices B j ∈ GL(R d ) are determined by the frame construction and δ > 0 globally stands for the sampling density.

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Gabor Expansions of Signals: Computational Aspects and Open Questions

Feichtinger

2019

Applied and Numerical Harmonic Analysis

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Gabor Frames: Characterizations and Coarse Structure

Abstract: This chapter offers a systematic and streamlined exposition of the most important characterizations of Gabor frames over a lattice. arXiv:1803.05271v1 [math.FA]

Cited by 9 publications

References 48 publications

Gabor frame bound optimizations

Gabor frame bound optimizations

Design and properties of wave packet smoothness spaces

Gabor Expansions of Signals: Computational Aspects and Open Questions

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