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On continuous weaving frames

On continuous weaving frames Abstract Two discrete frames { ϕ i } i ∈ I ${\{\phi_{i}\}_{i\in I}}$ and { ψ i } i ∈ I ${\{\psi_{i}\}_{i\in I}}$ for a separable Hilbert space ℍ ${\mathbb{H}}$ are said to be woven if there are universal positive constants A and B such that for every subset σ ⊂ I ${\sigma\subset I}$ , the family { ϕ i } i ∈ σ ∪ { ψ i } i ∈ σ c ${\{\phi_{i}\}_{i\in\sigma}\cup\{\psi_{i}\}_{i\in\sigma^{c}}}$ is a frame for ℍ ${\mathbb{H}}$ with lower and upper frame bounds A and B , respectively. Weaving frames are powerful tools in pre-processing signals and distributed data processing. Motivated by the recent work of Bemrose–Casazza–Gröchenig–Lammers–Lynch and Casazza–Lynch on weaving frames for separable Hilbert spaces, we study continuous weaving frames for Hilbert spaces with respect to a measure space. In this paper, first we present two necessary conditions in terms of frame bounds for continuous weaving frames. A sufficient condition for continuous weaving frames is given. We provide an estimate of a series associated with the metric operators of continuous weaving frames. Finally, two Paley–Wiener-type perturbation results for continuous weaving frames are given. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Pure and Applied Mathematics de Gruyter

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Publisher
de Gruyter
Copyright
Copyright © 2017 by the
ISSN
1867-1152
eISSN
1869-6090
DOI
10.1515/apam-2015-0077
Publisher site
See Article on Publisher Site

Abstract

Abstract Two discrete frames { ϕ i } i ∈ I ${\{\phi_{i}\}_{i\in I}}$ and { ψ i } i ∈ I ${\{\psi_{i}\}_{i\in I}}$ for a separable Hilbert space ℍ ${\mathbb{H}}$ are said to be woven if there are universal positive constants A and B such that for every subset σ ⊂ I ${\sigma\subset I}$ , the family { ϕ i } i ∈ σ ∪ { ψ i } i ∈ σ c ${\{\phi_{i}\}_{i\in\sigma}\cup\{\psi_{i}\}_{i\in\sigma^{c}}}$ is a frame for ℍ ${\mathbb{H}}$ with lower and upper frame bounds A and B , respectively. Weaving frames are powerful tools in pre-processing signals and distributed data processing. Motivated by the recent work of Bemrose–Casazza–Gröchenig–Lammers–Lynch and Casazza–Lynch on weaving frames for separable Hilbert spaces, we study continuous weaving frames for Hilbert spaces with respect to a measure space. In this paper, first we present two necessary conditions in terms of frame bounds for continuous weaving frames. A sufficient condition for continuous weaving frames is given. We provide an estimate of a series associated with the metric operators of continuous weaving frames. Finally, two Paley–Wiener-type perturbation results for continuous weaving frames are given.

Journal

Advances in Pure and Applied Mathematicsde Gruyter

Published: Jan 1, 2017

References