On \(n\)-term approximation with positive coefficients
type: Item
instance of: scholarly article
title: On \(n\)-term approximation with positive coefficients
zbMATH Open document ID: 1156.41008
DOI: 10.1134/S0001434607090052
author: E. D. Livshits
published in: Mathematical Notes
publication date: 2008-09-03T00:00:00Z
review text: The paper considers the Pure Greedy Algorithm (PGA) and the Weak Greedy Algorithm (WGA), and presents algorithms for constructing \(n\)-terms approximations with nonnegative coefficients. The convergence theorem is proved for a~``positive'' analog of the PGA. A condition on the sequence of weakness coefficients that is sufficient for the convergence of the Positive WGA is established as follows: Suppose that we are given a weakness sequence \(\tau= \{T_m\}^\infty_{m=1} \subset [0,1]\}\) such that \(\sum t_m/m <\infty\). Then there is a~positive dictionary \({\mathcal D}\) of a~real separable Hilbert space~\(H\) and a~target function \(f_0\in H\) such that there exists an~implementation of the PWGA \(\{G^{\text{WGA}} _m +(f_0,{\mathcal D},\tau)\}^\infty_{m=0}\) that fails to converge to~\(f_0\) as \(m\to \infty\). This condition is also necessary for the class of monotone sequences.
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A30
zbMATH DE Number: 5319843
zbMATH Keywords: polynomial approximation
zbMATH Keywords: greedy algorithm
zbMATH Keywords: approximation theory
zbMATH Keywords: positive dictionary
zbMATH Keywords: redundant system
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5319843