The theory of reproducing systems on locally compact abelian groups

Kutyniok, Gitta; Labate, Demetrio

doi:10.4064/cm106-2-3

Cited by 29 publications

(51 citation statements)

References 21 publications

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“…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%

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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“…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%

Co-compact Gabor Systems on Locally Compact Abelian Groups

Jakobsen

Lemvig

2015

J Fourier Anal Appl

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“…We refer the reader to the classical texts [12,14,27] and the recent book [38] for introductions to the specific cases of Gabor, wavelet, shearlet and wave packet analysis. In [36], Kutyniok and Labate generalized the results of Hernández, Labate, and Weiss to generalized shift invariant systems ∪ j∈J {T γ g j } γ∈Γ j in L 2 (G), where G is a second countable locally compact abelian group and Γ j is a family of uniform lattices (i.e., Γ j is a discrete subgroup and the quotient group G/Γ j is compact) indexed by a countable set J. The main goal of the present paper is to develop the corresponding theory for semi-continuous and continuous frames in L 2 (G).…”

Section: R\{0}mentioning

confidence: 99%

“…The characterization results in [30,36] rely on a technical condition on the generators and the translation lattices, the so-called local integrability condition. This condition is straightforward to formulate for generalized translation invariant systems, however, we will replace it by a strictly weaker condition, termed α local integrability condition.…”

Section: R\{0}mentioning

confidence: 99%

“…Finally, as Kutyniok and Labate [36] restrict their attention to Parseval frames, there are currently no characterization results available for dual (discrete) frames in the setting of locally compact abelian groups. Hence, one additional objective of this paper is to prove characterizing equations for dual generalized translation invariant frames to remedy this situation.…”

Section: R\{0}mentioning

confidence: 99%

“…When each P j is countable and each Γ j is a uniform lattice, i.e., a discrete, co-compact subgroup, we recover the generalized shift invariant (GSI) systems considered in [36]. However, we note that there exist locally compact abelian groups that do not contain any uniform lattices.…”

Section: Definition Of Generalized Translation Invariant Systemsmentioning

confidence: 99%

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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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Spline function generating Gabor systems on the half real line

Yang

2023

Math Methods in App Sciences

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Recently, Gabor analysis on locally compact abelian (LCA) groups has interested some mathematicians. The half real line R + = (0, ∞) is an LCA group under multiplication and the usual topology. This paper addresses spline Gabor frames for L 2 (R + , d𝜇), where 𝜇 is the corresponding Haar measure. We introduce the concept of spline functions on R + by 𝜇-convolution and estimate their Gabor frame sets, that is, lattice sets such that spline generating Gabor systems are frames for L 2 (R + , d𝜇). For an arbitrary spline Gabor frame with special lattices, we present its one dual Gabor frame window, which has the same smoothness as the initial window function. For a class of special spline Gabor Bessel sequences, we prove that they can be extended to a tight Gabor frame by adding a new window function, which has compact support and same smoothness as the initial windows. And we also demonstrate that two spline Gabor Bessel sequences can always be extended to a pair of dual Gabor frames with the adding window functions being compactly supported and having the same smoothness as the initial windows.

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The theory of reproducing systems on locally compact abelian groups

Abstract: Abstract.A reproducing system is a countable collection of functions {φ j

Cited by 29 publications

References 21 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

Spline function generating Gabor systems on the half real line

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