General Haar systems and greedy approximation

Anna Kamont¹

¹ Institute of Mathematics Polish Academy of Sciences Abrahama 18 81-825 Sopot, Poland

Studia Mathematica, Tome 145 (2001) no. 2, pp. 165-184

Résumé

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 $, $p \not =2$.

Détail

DOI : 10.4064/sm145-2-5

Keywords: each general haar system permutatively equivalent infty subsequence classical dyadic haar system consequence each general haar system greedy basis infty addition example general haar system whose tensor products greedy bases each infty mathbb contrast where has shown tensor products dyadic haar system greedy bases infty geq note above mentioned general haar system permutatively equivalent whole dyadic haar system infty