Wavelets in weighted norm spaces

TL;DR

This paper characterizes the weight functions for which Haar wavelets form an unconditional basis in weighted $L^p$ spaces, revealing that weights with strong zeros, like $|x|^r$, can still support such bases, expanding the known class of weights.

Contribution

It provides a complete characterization of weights for Haar wavelet bases in weighted $L^p$ spaces, especially including weights with strong zeros at the origin.

Findings

Higher rank Haar wavelets are unconditional bases in $L^p( ,w)$ for weights like $|x|^r$, $r>p-1$.

The class of weights supporting unconditional bases is broader than previously thought.

Weights with strong zeros at the origin can still support Haar wavelet bases.

Abstract

We give a complete characterization of the classes of weight functions for which the Haar wavelet system for $m$-dilations, $m= 2,3,\ldots$ is an unconditional basis in $L^p(\mathbb{R},w)$. Particulary it follows that higher rank Haar wavelets are unconditional bases in the weighted norm spaces $L^p(\mathbb{R},w)$, where $w(x) = |x|^{r}, r>p-1$. These weights can have very strong zeros at the origin. Which shows that the class of weight functions for which higher rank Haar wavelets are unconditional bases is much richer than it was supposed. One of main purposes of our study is to show that weights with strong zeros should be considered if somebody is studying basis properties of a given wavelet system in a weighted norm space.