Quantization of TF lattice-invariant operators on elementary LCA groups

Feichtinger, Hans G.; Kozek, W.

doi:10.1007/978-1-4612-2016-9_8

Cited by 106 publications

(173 citation statements)

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“…By [17,Corollary 7.6.6] we have that g, h ∈ S 0 (G) implies (x, ω) → g, E ω T x h ∈ S 0 (G × G). If we restrict this mapping to Γ ⊥ × Λ ⊥ ⊂ G × G and use that S 0 is continuously embedded into L 1 , we find that (5.8) is satisfied.…”

Section: The Janssen Representations Of the Frame Operatormentioning

confidence: 90%

“…In [19] Feichtinger and Luef give a detailed answer to when (5.10) holds in the setting of R n , see also [17,21] for related results. The FIGA was first proved by Rieffel [38] for generators g, h in the Schwartz-Bruhat space S(G).…”

Section: The Janssen Representations Of the Frame Operatormentioning

confidence: 99%

“…Feichtinger and Kozek [17] proved the Wexler-Raz biorthogonality relations for Gabor systems with translation and modulation along uniform lattices on elementary LCA groups, i.e., G = R n × T ℓ × Z k × F m , where F m is a finite group. For a proof in the discrete and finite setting and on the real line we refer to the original papers [45] and [31].…”

Section: Wexler-raz Biorthogonality Relationsmentioning

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: The Janssen Representations Of the Frame Operatormentioning

confidence: 90%

Section: The Janssen Representations Of the Frame Operatormentioning

confidence: 99%

Section: Wexler-raz Biorthogonality Relationsmentioning

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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“…An alternative approach that also works for non-product time-frequency lattices was developed in [11,14].…”

Section: Remark 45mentioning

confidence: 99%

Operators commuting with a discrete subgroup of translations

et al. 2006

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“…In order to derive the claimed convergence statement it now makes sense to describe both G k and G 0 in terms of a (double) sum of rank one operators (cf. [13]), which we can then split into two parts, the sum over F and (…”

Section: Gabor Multipliers and Their Continuitymentioning

confidence: 99%

Gabor multipliers with varying lattices

Feichtinger

2003

SPIE Proceedings

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In the early days of Gabor analysis it was a common to say that Gabor expansions of signals are interesting due to the natural interpretation of Gabor coefficients, but unfortunately the computation of Gabor coefficients is costly. Nowadays a large variety of efficient numerical algorithms exists (see [28] for a survey) and it has been recognized that stable and robust Gabor expansions can be achieved at low redundancy, e.g., by using a Gaussian atom and any time-frequency lattice of the formwith ab < 1. Consequently Gabor multipliers, i.e., linear operators obtained by applying a pointwise multiplication of the Gabor coefficients, become an important class of time-variant filters.It is the purpose of this paper to describe the fact that -provided one uses Gabor atoms from a suitable subspace-one has the expected continuous dependence of Gabor multipliers on the ingredients. In particular, we will provide new results which show that a small change of lattice parameters implies only a small change of the corresponding Gabor multiplier (e.g., in the Hilbert-Schmidt norm).

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Quantization of TF lattice-invariant operators on elementary LCA groups

Cited by 106 publications

References 0 publications

Co-compact Gabor Systems on Locally Compact Abelian Groups

Co-compact Gabor Systems on Locally Compact Abelian Groups

Operators commuting with a discrete subgroup of translations

Gabor multipliers with varying lattices

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