The Zak transform and the structure of spaces invariant by the action of an LCA group

Barbieri, Davide; Hernández, Eugenio; Paternostro, Victoria

doi:10.1016/j.jfa.2015.06.009

Cited by 50 publications

(65 citation statements)

References 39 publications

(125 reference statements)

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“…A very general version of this result was obtained independently and concurrently in [28]. Theorem 3.1 is closely related to the theory of translation invariant subspaces which very recently has been studied in [4,28] using Zak transform methods (cf. Section 4.1).…”

Section: G) With Bounds a And B (Or A Bessel System With Bound B)mentioning

confidence: 91%

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: G) With Bounds a And B (Or A Bessel System With Bound B)mentioning

confidence: 91%

Co-compact Gabor Systems on Locally Compact Abelian Groups

Jakobsen

Lemvig

2015

J Fourier Anal Appl

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“…Also the Zak transform on semidirect product groups with the action of a lattice subgroup of the abelian factor in is a special case of our definition. In , the authors propose a more general representation theoretic approach. They consider a unitary representation

σ

of a discrete lca group

G

L^{2} false(X false)

and the transform

0true \sum_{g \in G} σ_{g} f (x) \bar{χ} (g)

x \in X

χ \in \hat{G}

for

f \in L^{2} false(X false)

.…”

Section: The Zak Transform For Abelian Actionsmentioning

confidence: 99%

The Zak transform on strongly proper G-spaces and its applications

Jüstel

2017

J. London Math. Soc.

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Abstract. The Zak transform on R d is an important tool in condensed matter physics, signal processing, time-frequency analysis, and harmonic analysis in general. This article introduces a generalization of the Zak transform to a class of locally compact G-spaces, where G is either a locally compact abelian or a second countable unimodular type I group. This framework unifies previously proposed generalizations of the Zak transform. It is shown that the Zak transform has invariance properties analog to the classic case and is a Hilbert space isomorphism between the space of L 2 -functions and a direct integral of Hilbert spaces that is explicitly determined via a Weil formula for G-spaces and a Poisson summation formula for compact subgroups. Some applications in physics are outlined.

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“…For a locally compact abelian (LCA) group G, a translation invariant space is defined to be a closed subspace of L 2 (G) that is invariant under translations by elements of a closed subgroup Γ of G. Translation invariant spaces in case of Γ closed, discrete and cocompact, called shift invariant spaces, have been studied in [4,5,12,13,14,17], and extended to the case of Γ closed and cocompact (but not necessarily discrete) in [3] (see also [8,9]). Recently, translation invariant spaces have been generalized in [2] to the case when Γ is closed (not necessarily discrete or cocompact), see also [11]. Another spaces, which are effective tools in Gabor theory, are spaces invariant under modulations.…”

Section: Introductionmentioning

confidence: 99%

“…The basic idea is the fact that the image of a modulation invariant subspace of L 2 (G) under the Fourier transform is a translation invariant subspace of L 2 ( G). We use this fact to transform modulation invariant spaces to translation invariant spaces and then we follow the ideas in [3,2]. By transforming L 2 (G) into a vector valued space, in such a way that modulations by a closed subgroup of G become multiplications by a nice family of functions, we characterize modulation invariant spaces in terms of range functions.…”

Section: Introductionmentioning

confidence: 99%

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Modulation Preserving Operators on Locally Compact Abelian Groups

Mortazavizadeh

Tousi

2019

Bull. Malays. Math. Sci. Soc.

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We define and investigate modulation invariant spaces on a locally compact abelian group G with respect to a closed subgroup of the dual group G. Using a range function approach, we establish a characterization of modulation invariant spaces. Finally, we define a metric on the collection of all modulation invariant spaces and study some topological properties of the metric space.2010 Mathematics Subject Classification. Primary 47A15 ; Secondary 42B99, 22B99.

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The Zak transform and the structure of spaces invariant by the action of an LCA group

Cited by 50 publications

References 39 publications

Co-compact Gabor Systems on Locally Compact Abelian Groups

Co-compact Gabor Systems on Locally Compact Abelian Groups

The Zak transform on strongly proper G-spaces and its applications

Modulation Preserving Operators on Locally Compact Abelian Groups

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