## Greedy Approximation and Multivariate Haar System

- Dec. 4, 2001, 3:30 p.m. LeConte 312
- Dec. 11, 2001, 3:30 p.m. LeConte 312

## Abstract

The following two problems are discussed.

- We give a necessary and sufficient condition for convergence of a weak thresholding greedy algorithm with respect to an unconditional basis in a Banach space. As an example, we discuss this condition in the case of multi-variate (i. e., tensor product) Haar system in $L^p[0,1]^d,\;1<p<\infty,\;d\geq 2$.
- It is well known that the thresholding greedy algorithm with regard to the tensor product Haar system does not provide near-best $m$-term approximation for a general function in $L^p[0,1]^d\;(p\neq 2,\;d\geq 2)$. However, such algorithm still can provide the near best approximation at least for
**some functions**in $L^p[0,1]^d$. In the second part of the talk, we address the question of finding**greedy configurations**for the tensor product Haar system, i. e. such subsystems of the product Haar system for which the thresholding algorithm provides near-best $m$-term approximation. In this part, we focus our attention on the two-variate case.