Density and duality theorems for regular Gabor frames

Jakobsen, Mads Sielemann; Lemvig, Jakob

doi:10.1016/j.jfa.2015.10.007

Cited by 39 publications

(53 citation statements)

References 31 publications

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“…It is possible to consider Gabor systems over closed subgroups ∆ of G × G that are not discrete. This leads to the notion of continuous Gabor systems and frames, see [29]. In our case, we will eventually consider ∆ to be of the form Λ × Λ ⊥ , with Λ a closed subgroup of G. It turns out that for frames to exist over such a closed subgroup of G × G, Λ must be a lattice in G [29, Corollary 5.8], so we will not lose anything interesting by assuming that ∆ is discrete.…”

Section: Time-frequency Analysis and Heisenberg Modulesmentioning

confidence: 99%

The Balian–Low theorem for locally compact abelian groups and vector bundles

Enstad

2020

Journal de Mathématiques Pures et Appliquées

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Let Λ be a lattice in a second countable, locally compact abelian group G with annihilator Λ ⊥ ⊆ G. We investigate the validity of the following statement: For every η in the Feichtinger algebra S 0 (G), the Gabor system {Mτ T λ η} λ∈Λ,τ ∈Λ ⊥ is not a frame for L 2 (G). When G = R and Λ = αZ, this statement is a variant of the Balian-Low theorem. Extending a result of R. Balan, we show that whether the statement generalizes to (G, Λ) is equivalent to the nontriviality of a certain vector bundle over the compact space (G/Λ) × ( G/Λ ⊥ ). We prove this equivalence using a connection between Gabor frames and Heisenberg modules. More specifically, we show that the Zak transform can be viewed as an isomorphism of certain Hilbert C * -modules. As an application, we prove a new Balian-Low theorem for the group R × Qp, where Qp denotes the p-adic numbers.

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Section: Time-frequency Analysis and Heisenberg Modulesmentioning

confidence: 99%

The Balian–Low theorem for locally compact abelian groups and vector bundles

Enstad

2020

Journal de Mathématiques Pures et Appliquées

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“…The main goal of this section is to give a new proof for Theorem 9, which connects finite redundancy with the structure of the measure space, a result that has already been stated in [23, Theorem 2], [9, Theorem 2.2] and [24,Proposition 3.3]. We thereby use the property that Ran C Ψ forms a RKHS.…”

Section: Frames and Lower Semi-framesmentioning

confidence: 98%

Frames, their relatives and reproducing kernel Hilbert spaces

Speckbacher

Balázs

2019

J. Phys. A: Math. Theor.

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This paper considers different facets of the interplay between reproducing kernel Hilbert spaces (RKHS) and stable analysis/synthesis processes: First, we analyze the structure of the reproducing kernel of a RKHS using frames and reproducing pairs. Second, we present a new approach to prove the result that finite redundancy of a continuous frame implies atomic structure of the underlying measure space. Our proof uses the RKHS structure of the range of the analysis operator. This in turn implies that all the attempts to extend the notion of Riesz basis to general measure spaces are fruitless since every such family can be identified with a discrete Riesz basis. Finally, we show how the range of the analysis operators of a reproducing pair can be equipped with a RKHS structure.MSC2010: 42C15, 46E22 (primary), 47B32 (secondary)

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“…Let p be a seminorm defining the topology of D. where we have used the seminorms defined in (4) and in (26). This inequality easily implies that Ran C ω,θ is closed in V θ .…”

Section: Compatible Pairsmentioning

confidence: 99%

Distribution Frames and Bases

Trapani

Triolo

Tschinke

2019

J Fourier Anal Appl

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In this paper we will consider, in the abstract setting of rigged Hilbert spaces, distribution valued functions and we will investigate conditions for them to constitute a "continuous basis" for the smallest space D of a rigged Hilbert space. This analysis requires suitable extensions of familiar notions as those of frames, Riesz bases and orthonormal bases. A motivation for this study comes from the Gel'fand-Maurin theorem which states, under certain conditions, the existence of a family of generalized eigenvectors of an essentially self-adjoint operator on a domain D which acts like an orthonormal basis of the Hilbert space H. The corresponding object will be called here a Gel'fand distribution basis. The main results are obtained in terms of properties of a conveniently defined synthesis operator.

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Density and duality theorems for regular Gabor frames

Cited by 39 publications

References 31 publications

The Balian–Low theorem for locally compact abelian groups and vector bundles

The Balian–Low theorem for locally compact abelian groups and vector bundles

Frames, their relatives and reproducing kernel Hilbert spaces

Distribution Frames and Bases

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