Duality Principles in Frame Theory

Casazza, Peter G.; Kutyniok, Gitta; Lammers, Mark

doi:10.1007/s00041-004-3024-7

Cited by 64 publications

(80 citation statements)

References 15 publications

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“…We will study properties of dual g-frames and canonical dual g-frames. This is a general version of duality principle for g-frames which follows from the Casazza duality relations [4].…”

Section: Duality Properties Of the G-r-dual Sequencementioning

confidence: 95%

“…It can be viewed as a general version of [4,Proposition 13]. (i) Λ = {Λ i } i∈I is a g-frame sequence with respect to {V i } i∈I with g-frame bounds A, B.…”

Section: Characterizations Of Equivalence Of the G-r-dual Sequencementioning

confidence: 99%

“…Casazza et al in [4] introduced a general approach to derive duality principles in abstract frame theory. For each sequence in a separable Hilbert space they defined a Riesz-dual sequence dependent only on two orthonormal bases.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Duality and biorthogonality for g-frames in Hilbert spaces

Enayati¹

2017

J. Math. Computer Sci.

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“…We will study properties of dual g-frames and canonical dual g-frames. This is a general version of duality principle for g-frames which follows from the Casazza duality relations [4].…”

Section: Duality Properties Of the G-r-dual Sequencementioning

confidence: 95%

“…It can be viewed as a general version of [4,Proposition 13]. (i) Λ = {Λ i } i∈I is a g-frame sequence with respect to {V i } i∈I with g-frame bounds A, B.…”

Section: Characterizations Of Equivalence Of the G-r-dual Sequencementioning

confidence: 99%

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Duality and biorthogonality for g-frames in Hilbert spaces

Enayati¹

2017

J. Math. Computer Sci.

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“…We mention that duality results in the discrete case have been generalized in other directions; we refer the reader to [4,10,11,22,35] and the references therein. Generalizations of the density theorem also exist; in particular, Ramanathan and Steger [32] obtained density results for G (g, ∆) in L 2 (R n ), where ∆ is a discrete set, but not necessarily a subgroup.…”

Section: Introductionmentioning

confidence: 99%

Density and duality theorems for regular Gabor frames

Jakobsen

Lemvig

2016

Journal of Functional Analysis

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We investigate Gabor frames on locally compact abelian groups with time-frequency shifts along non-separable, closed subgroups of the phase space. Density theorems in Gabor analysis state necessary conditions for a Gabor system to be a frame or a Riesz basis, formulated only in terms of the index subgroup. In the classical results the subgroup is assumed to be discrete. We prove density theorems for general closed subgroups of the phase space, where the necessary conditions are given in terms of the "size" of the subgroup. From these density results we are able to extend the classical Wexler-Raz biorthogonal relations and the duality principle in Gabor analysis to Gabor systems with time-frequency shifts along non-separable, closed subgroups of the phase space. Even in the euclidean setting, our results are new.

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“…4 For a Bessel sequence in a separable Hilbert space an associated sequence, the so-called R-dual sequence, has been introduced. The relation between those two sequences turned out to be an important tool for deriving duality principles in abstract frame theory.…”

mentioning

confidence: 99%

Duality principles, localization of frames, and Gabor theory

2005

Self Cite

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The theory of localized frames is a recently introduced concept with broad implications to frame theory in general, as well as to the special cases of Gabor and wavelet frames. Using the new notion of a R-dual sequence associated with a Bessel sequence, we derive several duality principles concerning localization in abstract frame theory. As applications of our results we prove a duality principle of localization of Gabor systems in the spirit of the Ron-Shen duality principle, and obtain a Janssen representation for general frame operators.

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Duality Principles in Frame Theory

Cited by 64 publications

References 15 publications

Duality and biorthogonality for g-frames in Hilbert spaces

Duality and biorthogonality for g-frames in Hilbert spaces

Density and duality theorems for regular Gabor frames

Duality principles, localization of frames, and Gabor theory

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