Weak thresholding greedy algorithms in Banach spaces
type: Item
instance of: scholarly article
title: Weak thresholding greedy algorithms in Banach spaces
zbMATH Open document ID: 1267.46026
author: D. Kutzarova
author: Th. Schlumprecht
author: S. J. Dilworth
author: P. Wojtaszczyk
published in: Journal of Functional Analysis
publication date: 2013-01-17T00:00:00Z
review text: The paper is motivated by \textit{S. Gogyan}'s weak thresholding version of the thresholding greedy algorithm in \(L^1 (0,1)\) with regard to the Haar system [J. Approx. Theory 161, No. 1, 49--64 (2009; Zbl 1177.41032)]. The authors extract the basic features of Gogyan's weak thresholding greedy algorithm into a new definition of branch greedy algorithms with respect to Markushevich bases in general Banach spaces. The relationship between convergence of the algorithm and uniform boundedness of the approximants is studied: these two properties are not the same in general, but are equivalent for a natural subclass of branch greedy algorithms. It is shown that, if there is a branch greedy algorithm for the system which gives the best \(n\)-term approximation up to a multiplicative constant, then the system is already greedy. Similar results are proved for branch almost greedy systems in arbitrary spaces and for branch semi-greedy Schauder bases in spaces of finite cotype.
reviewed by: Vladimir Kadets
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 6126534
zbMATH Keywords: greedy approximation
zbMATH Keywords: Banach spaces
zbMATH Keywords: thresholding greedy algorithm
zbMATH Keywords: weak thresholding
zbMATH Keywords: branch greedy algorithm
MaRDI profile type: Publication
full work available at URL: j.jfa.2012.09.011
OpenAlex ID: W2027199502
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DOI: 10.1016/J.JFA.2012.09.011
description: scientific article; zbMATH DE number 6126534