Square Integrable Representations, von Neumann Algebras and An Application to Gabor Analysis

Bekka, Bachir

doi:10.1007/s00041-004-3036-3

Cited by 45 publications

(70 citation statements)

References 37 publications

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“…Specifically, Bekka proved the following, cf. [18,Theorem 4]; note that this is only a special case of the more general results that are derived in that article. 4 Theorem 11 (Existence of Lattice Gabor Frames).…”

Section: Existence For General Latticesmentioning

confidence: 83%

“…However, it appears that part (a) of the Density Theorem was only established in its full generality recently, by Bekka [18]. Moreover, Bekka provided a positive answer to the existence question for arbitrary lattices.…”

Section: Parts (B) and (C) Of The Density Theorem Are Immediate Consementioning

confidence: 99%

“…Thus, it is quite interesting that Bekka proved in [18] that for any lattice Λ with 0 < vol(Λ) ≤ 1 there will exist some window g such that G(g, Λ) will be a frame. Specifically, Bekka proved the following, cf.…”

Section: Existence For General Latticesmentioning

confidence: 99%

See 2 more Smart Citations

History and Evolution of the Density Theorem for Gabor Frames

Heil

2007

J Fourier Anal Appl

176

137

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ABSTRACT. The Density Theorem for Gabor Frames is one of the fundamental results of time-frequency analysis. This expository survey attempts to reconstruct the long and very involved history of this theorem and to present its context and evolution, from the onedimensional rectangular lattice setting, to arbitrary lattices in higher dimensions, to irregular Gabor frames, and most recently beyond the setting of Gabor frames to abstract localized frames. Related fundamental principles in Gabor analysis are also surveyed, including the Wexler-Raz biorthogonality relations, the Duality Principle, the Balian-Low Theorem, the Walnut and Janssen representations, and the Homogeneous Approximation Property. An extended bibliography is included.

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Section: Existence For General Latticesmentioning

confidence: 83%

Section: Parts (B) and (C) Of The Density Theorem Are Immediate Consementioning

confidence: 99%

See 1 more Smart Citation

History and Evolution of the Density Theorem for Gabor Frames

Heil

2007

J Fourier Anal Appl

176

137

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“…For a full-rank lattice ∆ in R 2n , Bekka [3] proved (using von Neumann algebra techniques) that there exists g ∈ L 2 (R n ) so that G (g, ∆) is a frame if, and only if, there exists g ∈ L 2 (R n ) so that G (g, ∆) is total, i.e., the linear span of the functions in G (g, ∆) is dense in L 2 (R n ). This equivalence is not true for non-discrete Gabor systems, e.g., take ∆ = R n × {0} n .…”

Section: The Next Results Relate the Norm Of A Gabor Frame Generator mentioning

confidence: 99%

Density and duality theorems for regular Gabor frames

Jakobsen

Lemvig

2016

Journal of Functional Analysis

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We investigate Gabor frames on locally compact abelian groups with time-frequency shifts along non-separable, closed subgroups of the phase space. Density theorems in Gabor analysis state necessary conditions for a Gabor system to be a frame or a Riesz basis, formulated only in terms of the index subgroup. In the classical results the subgroup is assumed to be discrete. We prove density theorems for general closed subgroups of the phase space, where the necessary conditions are given in terms of the "size" of the subgroup. From these density results we are able to extend the classical Wexler-Raz biorthogonal relations and the duality principle in Gabor analysis to Gabor systems with time-frequency shifts along non-separable, closed subgroups of the phase space. Even in the euclidean setting, our results are new.

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“…The following is [7,Thm. 4], and is only a special case of the more general results obtained in that paper (Bekka himself attributes this result to Feichtinger and Kozek [20], but while as we have mentioned that paper does contain many results for Gabor systems on arbitrary lattices, it does not contain Theorem 6).…”

Section: Theorem 5 (Density Theorem For Latticesmentioning

confidence: 99%

The Density Theorem and the Homogeneous Approximation Property for Gabor Frames

Heil

Representations, Wavelets, and Frames

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Summary. The Density Theorem for Gabor Frames is a fundamental result in timefrequency analysis. Beginning with Baggett's proof that a rectangular lattice Gabor system {e 2πiβnt g(t − αk)} n,k∈Z must be incomplete in L 2 (R) whenever αβ > 1, the necessary conditions for a Gabor system to be complete, a frame, a Riesz basis, or a Riesz sequence have been extended to arbitrary lattices and beyond. The first partial proofs of the Density Theorem for irregular Gabor frames were given by Landau in 1993 and by Ramanathan and Steger in 1995. A key fact proved by Ramanathan and Steger is that irregular Gabor frames possess a certain Homogeneous Approximation Property (HAP), and that the Density Theorem is a consequence of this HAP. This chapter provides a brief history of the Density Theorem and a detailed account of the proofs of Ramanathan and Steger. Furthermore, we show that the techniques of Ramanathan and Steger can be used to give a full proof of a general version of Density Theorem for irregular Gabor frames in higher dimensions and with finitely many generators.

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Square Integrable Representations, von Neumann Algebras and An Application to Gabor Analysis

Cited by 45 publications

References 37 publications

History and Evolution of the Density Theorem for Gabor Frames

History and Evolution of the Density Theorem for Gabor Frames

Density and duality theorems for regular Gabor frames

The Density Theorem and the Homogeneous Approximation Property for Gabor Frames

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