Fourier-like frames on locally compact abelian groups

Christensen, Ole; Goh, Say Song

doi:10.1016/j.jat.2014.11.002

Cited by 31 publications

(40 citation statements)

References 10 publications

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“…, g r ∈ L 2 (G) be functions positive on X with support supp g i ⊂ X so that g i is constant on X for at least one index i. Following [13], the function on G defined by the r-fold convolution…”

Section: Characterizations Of Dual Gabor Framesmentioning

confidence: 99%

“…is called a weighted B-spline of order r. As shown in [13], the function W r is non-negative and satisfies a partition of unity condition up to a constant, say λ∈Λ W r (x − λ) = C r . Take g ∈ L 2 (G) so that…”

Section: Characterizations Of Dual Gabor Framesmentioning

confidence: 99%

“…For related work on locally compact (abelian) groups we refer to the recent papers [2,3,7,8,13,18,29,34] as well as the book [20] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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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: Characterizations Of Dual Gabor Framesmentioning

confidence: 99%

Section: Characterizations Of Dual Gabor Framesmentioning

confidence: 99%

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Co-compact Gabor Systems on Locally Compact Abelian Groups

Jakobsen

Lemvig

2015

J Fourier Anal Appl

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“…Proposition 3.7 is a generalization of the results in, e.g., [10] and [9], which state the corresponding result for GSI systems in the euclidean space and locally compact abelian groups, respectively. The result is as follows.…”

Section: On Sufficient Conditions and The Local Integrability Conditionsmentioning

confidence: 59%

“…We will focus on verifying the local integrability conditions and on the deriving the characterizing equations, but not on the related question of how to construct generators satisfying these equations. The recent work of Christensen and Goh [9] takes this more constructive approach for generalized shift invariant systems on locally compact abelian groups. Under certain assumptions, they explicitly construct dual GSI frames using variants of t α -equations, which are proved to be sufficient.…”

Section: Applications and Discussion Of The Characterization Resultsmentioning

confidence: 99%

Reproducing formulas for generalized translation invariant systems on locally compact abelian groups

Jakobsen

Lemvig

2016

Trans. Amer. Math. Soc.

104

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In this paper we connect the well established discrete frame theory of generalized shift invariant systems to a continuous frame theory. To do so, we let Γ j , j ∈ J, be a countable family of closed, co-compact subgroups of a second countable locally compact abelian group G and study systems of the form ∪ j∈J {g j,p (· − γ)} γ∈Γj,p∈Pj with generators g j,p in L 2 (G) and with each P j being a countable or an uncountable index set. We refer to systems of this form as generalized translation invariant (GTI) systems. Many of the familiar transforms, e.g., the wavelet, shearlet and Gabor transform, both their discrete and continuous variants, are GTI systems. Under a technical α local integrability condition (α-LIC) we characterize when GTI systems constitute tight and dual frames that yield reproducing formulas for L 2 (G). This generalizes results on generalized shift invariant systems, where each P j is assumed to be countable and each Γ j is a uniform lattice in G, to the case of uncountably many generators and (not necessarily discrete) closed, co-compact subgroups. Furthermore, even in the case of uniform lattices Γ j , our characterizations improve known results since the class of GTI systems satisfying the α-LIC is strictly larger than the class of GTI systems satisfying the previously used local integrability condition. As an application of our characterization results, we obtain new characterizations of translation invariant continuous frames and Gabor frames for L 2 (G). In addition, we will see that the admissibility conditions for the continuous and discrete wavelet and Gabor transform in L 2 (R n ) are special cases of the same general characterizing equations.

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Nonstationary Gabor frames for

Lian

2019

Math Methods in App Sciences

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Recently, continuous‐time nonstationary Gabor (NSG) frames were introduced in adaptive signal analysis. They allow for efficient reconstruction with flexible sampling and varying window functions. In this paper, we focus on the existence and construction of NSG frames in the discrete‐time setting. We provide existence results for painless NSG frames and for NSG frames with fast decaying window functions. We also construct NSG frames with noncompactly supported window functions from a known painless NSG frame. Some examples are provided to illustrate the general theory.

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Fourier-like frames on locally compact abelian groups

Cited by 31 publications

References 10 publications

Co-compact Gabor Systems on Locally Compact Abelian Groups

Co-compact Gabor Systems on Locally Compact Abelian Groups

Reproducing formulas for generalized translation invariant systems on locally compact abelian groups

Nonstationary Gabor frames for

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