A note on the solution of the Mexican hat problem

TL;DR

This paper provides a technical estimate crucial for establishing the completeness of the Mexican hat wavelet system in specific function spaces, advancing understanding in wavelet analysis.

Contribution

It offers a new technical estimate that supports the solution of the completeness problem for the Mexican hat wavelet system in certain $L^p$ and Hardy spaces.

Findings

Proves a key estimate for wavelet completeness

Extends results to $L^p$ spaces with $1<p<2$

Addresses Hardy space $H^p$ for $2/3 < p \,\leq 1$

Abstract

We prove a technical estimate needed in our recent solution of the completeness question for the non-orthogonal Mexican hat wavelet system, in $L^p$ for $1<p<2$ and in the Hardy space $H^p$ for $2/3 < p \leq 1$.