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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.

  1. 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 $L^p[0,1]^d,\;1<p<\infty,\;d\geq 2$.
  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.
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