Convergence of the weak dual greedy algorithm in \(L_{p}\)-spaces.
type: Item
instance of: scholarly article
title: Convergence of the weak dual greedy algorithm in \(L_{p}\)-spaces.
zbMATH Open document ID: 1041.46007
DOI: 10.1016/S0021-9045(03)00133-3
author: M. Ganichev
author: N. J. Kalton
published in: Journal of Approximation Theory
publication date: 2003-11-19T00:00:00Z
review text: A dictionary in a real Banach space \(X\) is a subset \(D\subseteq X\) which is (i) normalized: \(g\in D\) implies \(\| g\|=1\); (ii) symmetric: \(-D=D\); (iii) fundamental: \(X\) is the closed linear space generated by \(D\). The goal of the present paper is to construct algorithms giving, for each \(x\in X\), a sequence \(\sum_{k=1}^n t_k g_k\), with non-negative real numbers \(t_k\) and \(g_k\in D\), and which (as expected) converges to \(x\). The pure greedy algorithm (PGA) was introduced and studied by \textit{P. Huber} [Ann. Stat. 13, 435--525 (1985; Zbl 0595.62059)]; its convergence was proved by \textit{L. Jones} [Ann. Stat. 15, 880--882 (1987; Zbl 0664.62061)]. In Hilbert spaces, it was modified into the weak greedy algorithm (WGA) and its convergence was proved by \textit{L. Rejtö} and \textit{G. Walter} [Stochastic Anal. Appl. 10, No. 2, 213--222 (1992; Zbl 0763.62034)]. In the paper under review, the authors study a generalization, the weak dual greedy algorithm (WDGA), in Banach spaces with a Gâteaux differentiable norm. It was conjectured by \textit{V. Temlyakov} [Found. Comput. Math. 3, No. 1, 33--107 (2003; Zbl 1039.41012)] that (WDGA) converges when \(X\) is a uniformly smooth Banach space with power-type modulus of smoothness. The authors prove the conjecture when \(X\) is a subspace of a quotient of \(L^p\), \(1<p<+\infty\). For that purpose, they define a geometric property, called \(\Gamma\), in Banach spaces with a Gâteaux differentiable norm, and they prove that: (1) every subspace of a quotient of \(L^p\), \(1<p<+\infty\), has property \(\Gamma\) (the authors acknowledge that for \(p\geq 2\), the result was essentially proved by \textit{E. D. Livshits} [Math. Notes 73, 342--358 (2003)]; the key fact for their proof is the following elementary inequality: \(b| a+b|^{p-1} \text{sgn}(a+b) -b| a|^{p-1} \text{sgn}(a)\leq C_p(| a+b|^p-p\,b| a|^{p-1} \text{sgn}(a)-| a|^p)\); (2) in every Banach space \(X\) with Fréchet differentiable norm and which …
reviewed by: Daniel Li
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 2005938
zbMATH Keywords: dictionary
zbMATH Keywords: Fréchet differentiable norm
zbMATH Keywords: Gâteaux differentiable norm
zbMATH Keywords: greedy algorithm
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2005938