On Various R-duals and the Duality Principle

Stoeva, Diana T.; Christensen, Ole

doi:10.1007/s00020-016-2283-4

Cited by 10 publications

(4 citation statements)

References 18 publications

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“…The following central result of Gabor frames is due to the seminal work [10,20,28]. The possible extension of the duality principle from Gabor systems to other types of systems has been investigated in [1,4,5,12] and [16] as well as in the form of the theory of R-duality [7,9,30,31].…”

Section: Introductionmentioning

confidence: 99%

Gabor duality theory for Morita equivalent C∗-algebras

Austad

Jakobsen²,

Luef

2020

Int. J. Math.

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The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalence bimodules with some extra properties. For certain twisted group [Formula: see text]-algebras, the reformulation of the duality principle to the setting of Morita equivalence bimodules reduces to the well-known Gabor duality principle by localizing with respect to a trace. We may lift all results at the module level to matrix algebras and matrix modules, and in doing so, it is natural to introduce [Formula: see text]-matrix Gabor frames, which generalize multi-window super Gabor frames. We are also able to establish density theorems for module frames on equivalence bimodules, and these localize to density theorems for [Formula: see text]-matrix Gabor frames.

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Section: Introductionmentioning

confidence: 99%

Gabor duality theory for Morita equivalent C∗-algebras

Austad

Jakobsen²,

Luef

2020

Int. J. Math.

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“…Recently, the various generalizations of duality principles have been proposed. For example, duality principle for g-frames in Hilbert spaces [4,5,6], the duality principle for p-frames [7], and various R-duals [8,9]. In [10], the authors studied R-duals for the purpose of extending this to general sequences in arbitrary Banach spaces.…”

Section: Introductionmentioning

confidence: 99%

On various Riesz-dual sequences for Schauder frames

Neisi

Asgari

2020

Heliyon

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In this paper, we introduce various definitions of R-duals, to be called R-duals of type I, II, which leads to a generalization of the duality principle in Banach spaces. A basic problem of interest in connection with the study of R-duals in Banach spaces is that of characterizing those R-duals which can essentially be regarded as M-basis. We give some conditions under which an R-dual sequence to be an M-basis for .

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“…Stoeva and Christensen 11,12 introduced R‐duals of type II and III and proved that for a Gabor frame

scriptG false(g, a, b false)

\frac{1}{\sqrt{a b}} scriptG ((), g, \frac{1}{b}, \frac{1}{a})

is exactly one of its R‐duals of type III. Now many variants of R‐duals are proposed and investigated.…”

Section: Introductionmentioning

confidence: 99%

“…Stoeva and Christensen 11,12 introduced R-duals of type II and III and proved that for a Gabor frame (g, a, b),…”

Section: Introductionmentioning

confidence: 99%

Duality principles in Hilbert‐Schmidt frame theory

Dong

2020

Math Methods in App Sciences

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This paper addresses duality relations in HS‐frame theory. By introducing the notion of HS‐R‐dual sequence, some duality relations and related results are obtained. We prove that a sequence is an HS‐frame (HS‐frame sequence, HS‐Riesz basis) if and only if its HS‐R‐dual sequence is an HS‐Riesz sequence (HS‐frame sequence, HS‐Riesz basis) and characterize the (unitary) equivalence between two HS‐frames in terms of their HS‐R‐duals and transition matrices, respectively. We characterize HS‐R‐duals and prove that, given an HS‐frame, among all its dual HS‐frames, only the canonical dual admits minimal‐norm HS‐R‐dual. And using HS‐R‐duals, we also characterize dual HS‐frame pairs.

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On Various R-duals and the Duality Principle

Cited by 10 publications

References 18 publications

Gabor duality theory for Morita equivalent C∗-algebras

Gabor duality theory for Morita equivalent C∗-algebras

On various Riesz-dual sequences for Schauder frames

Duality principles in Hilbert‐Schmidt frame theory

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