Greedy approximation and the multivariate Haar system

A. Kamont ¹   ; V. N. Temlyakov ²

¹ Institute of Mathematics Polish Academy of Sciences Abrahama 18 81-825 Sopot, Poland
² Department of Mathematics University of South Carolina Columbia, SC 29208, U.S.A.

Studia Mathematica, Tome 161 (2004) no. 3, pp. 199-223

Résumé

We study nonlinear $m$-term approximation in a Banach space with regard to a basis. It is known that in the case of a greedy basis (like the Haar basis ${\mathcal H}$ in $L_p([0,1])$, $1 p \infty $) a greedy type algorithm realizes nearly best $m$-term approximation for any individual function. In this paper we generalize this result in two directions. First, instead of a greedy algorithm we consider a weak greedy algorithm. Second, we study in detail unconditional nongreedy bases (like the multivariate Haar basis ${\mathcal H}^d={\mathcal H}\times \mathinner {\ldotp \ldotp \ldotp }\times {\mathcal H}$ in $L_p([0,1]^d)$, $1 p \infty $, $p\not =2$). We prove some convergence results and also some results on convergence rate of weak type greedy algorithms. Our results are expressed in terms of properties of the basis with respect to a given weakness sequence.

Détail

DOI : 10.4064/sm161-3-1

Keywords: study nonlinear m term approximation banach space regard basis known greedy basis haar basis mathcal infty greedy type algorithm realizes nearly best m term approximation individual function paper generalize result directions first instead greedy algorithm consider weak greedy algorithm second study detail unconditional nongreedy bases multivariate haar basis mathcal mathcal times mathinner ldotp ldotp ldotp times mathcal infty prove convergence results results convergence rate weak type greedy algorithms results expressed terms properties basis respect given weakness sequence