General Haar systems and greedy approximation
Volume 145 / 2001
Abstract
We show that each general Haar system is permutatively equivalent in $L^p([0,1])$, $1 < p < \infty $, to a subsequence of the classical (i.e. dyadic) Haar system. As a consequence, each general Haar system is a greedy basis in $L^p([0,1])$, $1 < p < \infty $. In addition, we give an example of a general Haar system whose tensor products are greedy bases in each $L^p([0,1]^d)$, $1< p< \infty $, $d \in {\mathbb N}$. This is in contrast to [11], where it has been shown that the tensor products of the dyadic Haar system are not greedy bases in $L^p([0,1]^d)$ for $1< p< \infty $, $p \not =2$ and $d\geq 2$. We also note that the above-mentioned general Haar system is not permutatively equivalent to the whole dyadic Haar system in any $L^p([0,1])$, $1< p< \infty $, $p \not =2$.
