Wavelet frame bijectivity on Lebesgue and Hardy spaces

TL;DR

This paper establishes conditions under which wavelet frame series are bijective on Lebesgue, Hardy, and BMO spaces, demonstrating expansions with Mexican hat wavelets and advancing understanding of wavelet operator invertibility.

Contribution

It provides new sufficient conditions for wavelet frame bijectivity on various function spaces, including Hardy space, using novel frequency-domain estimates.

Findings

Functions in L^p, H^1, and BMO have wavelet expansions under these conditions.

Bijectivity of the wavelet frame operator on Hardy space is proved.

Mexican hat wavelet expansions are validated for these spaces.

Abstract

We prove a sufficient condition for frame-type wavelet series in $L^p$, the Hardy space $H^1$, and BMO. For example, functions in these spaces are shown to have expansions in terms of the Mexican hat wavelet, thus giving a strong answer to an old question of Meyer. Bijectivity of the wavelet frame operator acting on Hardy space is established with the help of new frequency-domain estimates on the Calder\'on-Zygmund constants of the frame kernel.