Convergence of the weak dual greedy algorithm in \(L_{p}\)-spaces.
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 has property \(\Gamma\), the (WDGA) converges for every \(x\in X\) and every dictionary. This result is new even when \(X\) is \(L^p\) itself and the dictionary \(D\) is the Haar basis. When \(D\) is the trigonometric system, see [\textit{S. Dilworth}, \textit{D. Kutzarova} and \textit{V. Temlyakov}, J. Fourier Anal. Appl. 8, No. 5, 489--505 (2002; Zbl 1025.41021)].
scientific article; zbMATH DE number 2005938