## General Haar systems as greedy bases

- Feb. 1, 2001
- 3:30 p.m.
- LeConte 312

## Abstract

By a general Haar system corresponding to a dense sequence $\cal T$ of points in $[0,1]$ we mean a sequence of orthonormal functions, which are constant on intervals generated by the points from $\cal T$. We show that each general Haar system is permutatively equivalent in $L^p([0,1])$, $1 < p < \infty$, to some subsequence of the dyadic Haar system, and, as a consequence, each general Haar system is a greedy basis in $L^p([0,1])$. Moreover, we show an example of a general Haar system such that its tensor products are greedy bases in each $L^p([0,1]^d)$, $1 < p < \infty$, $d \in N$ - in contrast, recall that the basis consisting of tensor products of the dyadic Haar system is not greedy in $L^p([0,1]^d)$ for $d \geq 2$ and $p \neq 2$.