Optimal dual frames for erasures
type: Item
instance of: scholarly article
title: Optimal dual frames for erasures
zbMATH Open document ID: 1181.42034
author: Jerry Lopez
author: Deguang Han
published in: Linear Algebra and its Applications
publication date: 2009-12-18T00:00:00Z
review text: Let \(X=\{x_1,\dots,x_n\}\) be a frame for a \(k\)-dimensional Hilbert space \(H\), where \(n>k\). In coding theory, a signal \(x\in H\) is encoded as \(\Theta(x) = (\langle x, x_j \rangle)_{j=1}^n\). Then \(\Theta(x)\) is sent to a receiver for decoding to reconstruct \(x\). But some coefficients of \(\Theta(x)\) may be lost in the transmission process. The authors reconstruct \(x\) using a dual frame that minimizes the error of reconstruction. Such a dual frame is called \textit{optimal dual frame for erasures}. It is shown that in general the canonical dual frame of \(X\) is not an optimal dual frame for erasures. Further, optimal dual frames are not necessarily unique. The authors present sufficient conditions (e.g., \(X\) is a tight uniform frame or a group frame) under which the canonical dual frame of \(X\) is the unique optimal dual frame for erasures.
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 94A29
Mathematics Subject Classification ID: 68P30
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 5650054
zbMATH Keywords: frame
zbMATH Keywords: dual frame
zbMATH Keywords: optimal dual frame for erasures
zbMATH Keywords: erasure
zbMATH Keywords: frame operator
zbMATH Keywords: canonical dual frame
zbMATH Keywords: reconstruction of signals
zbMATH Keywords: coding theory
zbMATH Keywords: tight uniform frame
zbMATH Keywords: group frame
MaRDI profile type: Publication
full work available at URL: j.laa.2009.08.031
OpenAlex ID: W2157312770
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DOI: 10.1016/J.LAA.2009.08.031
description: scientific article; zbMATH DE number 5650054