Meyer wavelets for rational dilations

TL;DR

This paper proves the existence of smooth, orthonormal Meyer wavelets with rational dilations in any dimension, extending previous one-dimensional results and establishing the limitations for well-localized wavelets.

Contribution

It demonstrates the existence of smooth Meyer wavelets for rational dilations in any dimension and shows such wavelets cannot exist for irrational dilations, extending prior one-dimensional findings.

Findings

Existence of smooth, orthonormal Meyer wavelets for rational dilations in any dimension.

Non-existence of well-localized orthogonal MRA wavelets for irrational dilations.

Extension of one-dimensional results to higher dimensions.

Abstract

We show the existence of smooth band-limited multiresolution analysis (MRA) for any expansive dilation with real entries in any spatial dimension. We then prove the existence of orthonormal Meyer wavelets, which have smooth and compactly supported Fourier transform, for any expansive dilation with rational entries and any spatial dimension. This extends one dimensional results of Auscher. In a converse direction, we show that well-localized orthogonal MRA wavelets, such as Meyer wavelets, can only exist for expansive dilations with rational entries. This shows the optimality of our existence result and extends one dimensional result of Lemari\'e-Rieusset.