Greedy approximation and the multivariate Haar system
Volume 161 / 2004
Abstract
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.
