History and Evolution of the Density Theorem for Gabor Frames

Heil, Christopher

doi:10.1007/s00041-006-6073-2

Cited by 176 publications

(141 citation statements)

References 185 publications

(314 reference statements)

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“…Landau [21] gave a density condition for Gabor frames with generators certain entire functions), partial progress towards a quantitative notion of redundancy has occurred for both lattice and general Gabor frames. Many works have connected essential features of the frames to quantities related to the density Λ of the associated set of time and frequency shifts (For an extensive account of the development of density theory for Gabor frames see [18] and references therein). As dynamic as these results were, they could not be used to show that the obvious choice for redundancy, namely the reciprocal of the density of Λ, satisfied any version of properties 3) or 4).…”

Section: Introductionmentioning

confidence: 99%

Redundancy for localized frames

2011

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Redundancy is the qualitative property which makes Hilbert space frames so useful in practice. However, developing a meaningful quantitative notion of redundancy for infinite frames has proven elusive. Though quantitative candidates for redundancy exist, the main open problem is whether a frame with redundancy greater than one contains a subframe with redundancy arbitrarily close to one. We will answer this question in the affirmative for 1 -localized frames. We then specialize our results to Gabor multi-frames with generators in M 1 (R d ), and Gabor molecules with envelopes in W (C, l 1 ). As a main tool in this work, we we answer a longstanding question in finite dimensional frame theory by showing there is a universal function g(x, y) so that for every > 0, every Parseval frame {f i } M i=1 for an N -dimensional Hilbert space H N has a subset of fewer than (1 + )N elements which is a frame for H N with lower frame bound g( , M N ). The result of this work is the first meaningful quantitative notion of redundancy for a large class of infinite frames.

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Section: Introductionmentioning

confidence: 99%

Redundancy for localized frames

2011

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“…One of the well known theorems about Gabor analysis is the density theorem which states the necessary condition of being basis and frames with respect to the modulation and translation parameters [15]. Let us…”

Section: Chaptermentioning

confidence: 99%

Gabor frames with trigonometric spline dual windows

Kim

2015

Asian-European J. Math.

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“…Generalizations of the density theorem also exist; in particular, Ramanathan and Steger [32] obtained density results for G (g, ∆) in L 2 (R n ), where ∆ is a discrete set, but not necessarily a subgroup. We refer the reader to the survey paper by Heil [24] for a detailed account of the history and evolution of density results in Gabor analysis. For an introduction to Gabor analysis and frame theory, we refer to [6,21].…”

Section: Introductionmentioning

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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History and Evolution of the Density Theorem for Gabor Frames

Cited by 176 publications

References 185 publications

Redundancy for localized frames

Redundancy for localized frames

Gabor frames with trigonometric spline dual windows

Density and duality theorems for regular Gabor frames

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