Generalized shift-invariant systems and approximately dual frames
type: Item
instance of: scholarly article
title: Generalized shift-invariant systems and approximately dual frames
zbMATH Open document ID: 1362.42060
DOI: 10.1215/20088752-3784315
author: Ana Benavente
author: Ole Christensen
author: María I. Zakowicz
published in: Annals of Functional Analysis
publication date: 2017-03-14T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.afa/1484363069
review text: Dual pairs of frames yield a procedure for obtaining a perfect reconstruction of elements in the underlying Hilbert space in terms of superpositions of the frame elements. However, practical constraints often force to consider nonperfect reconstructions. It is then important to study the deviation from the perfect reconstruction and to give tools to estimate it. The present paper deals especially with the case of generalized shift-invariant systems \(f_k (k\in\mathbb{N})\), \(g_k (k\in\mathbb{N})\) in \(L^2(\mathbb{R})\) and the estimation of the deviation is expressed by use of the operator norm \[ \left\| I-\sum_{k=1}^{+\infty}<.,g_k>f_k\right\|\;=\;\sup_{\|f\|=1}\left\|f-\sum_{k=1}^{+\infty}<f,g_k>f_k\right\|. \] It generalizes some known results and provides new ones. The paper is clearly written and structured. The list of references gives a very good idea of the current research in the domain and shows how the results of the paper can be integrated.
reviewed by: Françoise Bastin
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 46E40
zbMATH DE Number: 6694406
zbMATH Keywords: approximately dual frames
zbMATH Keywords: frames
zbMATH Keywords: generalized shift-invariant systems
MaRDI profile type: Publication
OpenAlex ID: W2572152798
description: scientific article; zbMATH DE number 6694406