Density of Gabor Frames

Christensen, Ole; Deng, Baiqiao; Heil, Christopher

doi:10.1006/acha.1999.0271

Cited by 129 publications

(122 citation statements)

References 9 publications

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“…ω ∈ G will generate a frame {T λ g} λ∈G with bounds A, B. For discrete (irregular) Gabor systems in L 2 (R n ) such questions are studied in [12]. On the other hand, totality in L 2 (G) of the set {T λ g} λ∈G is achievable for both discrete and non-discrete LCA groups G; e.g., take any g ∈ L 2 (G) withĝ(ω) = 0 for a.e.…”

Section: Proof the Implication (Ii)⇒(i) Is Trivial So We Only Have Tmentioning

confidence: 99%

Co-compact Gabor Systems on Locally Compact Abelian Groups

Jakobsen

Lemvig

2015

J Fourier Anal Appl

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In this work we extend classical structure and duality results in Gabor analysis on the euclidean space to the setting of second countable locally compact abelian (LCA) groups. We formulate the concept of rationally oversampling of Gabor systems in an LCA group and prove corresponding characterization results via the Zak transform. From these results we derive non-existence results for critically sampled continuous Gabor frames. We obtain general characterizations in time and in frequency domain of when two Gabor generators yield dual frames. Moreover, we prove the Walnut and Janssen representation of the Gabor frame operator and consider the Wexler-Raz biorthogonality relations for dual generators. Finally, we prove the duality principle for Gabor frames. Unlike most duality results on Gabor systems, we do not rely on the fact that the translation and modulation groups are discrete and co-compact subgroups. Our results only rely on the assumption that either one of the translation and modulation group (in some cases both) are co-compact subgroups of the time and frequency domain. This presentation offers a unified approach to the study of continuous and the discrete Gabor frames.

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Section: Proof the Implication (Ii)⇒(i) Is Trivial So We Only Have Tmentioning

confidence: 99%

Co-compact Gabor Systems on Locally Compact Abelian Groups

Jakobsen

Lemvig

2015

J Fourier Anal Appl

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“…The density result of Ramanthan and Steger [37] (as extended by Christensen, Deng and Heil [15]) is as follows:…”

Section: Theorem 32 (Homogeneous Approximation Propertymentioning

confidence: 99%

An Introduction to Irregular Weyl-Heisenberg Frames

Casazza¹

2004

Sampling, Wavelets, and Tomography

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“…[20], Theorems 1.1, 3.1. For a regular Gabor system {E mb T na g} m,n∈Z the latter assumption corresponds exactly to the condition ab ≤ 1.…”

Section: Irregular Gabor Systems and Open Problemsmentioning

confidence: 99%

“…Observe that V 0 will always be a pure subspace of L 2 (R): in [20] it is proved that translates of a single function never can constitute a frame for all of L 2 (R). A sufficient condition for {V j , φ} to be an FMRA was given in [4], Theorem 4.6: Note the similarity between Theorem 5.4 and the results for an MRA; cf.…”

Section: Frame Multiresolution Analysismentioning

confidence: 99%

Frames, Riesz bases, and discrete Gabor/wavelet expansions

Christensen

2001

Bull. Amer. Math. Soc.

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106

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Abstract. This paper is a survey of research in discrete expansions over the last 10 years, mainly of functions in L 2 (R). The concept of an orthonormal basis {fn}, allowing every function f ∈ L 2 (R) to be written f = cnfn for suitable coefficients {cn}, is well understood. In separable Hilbert spaces, a generalization known as frames exists, which still allows such a representation. However, the coefficients {cn} are not necessarily unique. We discuss the relationship between frames and Riesz bases, a subject where several new results have been proved over the last 10 years. Another central topic is the study of frames with additional structure, most important Gabor frames (consisting of modulated and translated versions of a single function) and wavelets (translated and dilated versions of one function). Along the way, we discuss some possible directions for future research.

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Density of Gabor Frames

Abstract: A Gabor system is a set of time-frequency shifts S(g, ) = {e

Cited by 129 publications

References 9 publications

Co-compact Gabor Systems on Locally Compact Abelian Groups

Co-compact Gabor Systems on Locally Compact Abelian Groups

An Introduction to Irregular Weyl-Heisenberg Frames

Frames, Riesz bases, and discrete Gabor/wavelet expansions

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