Meyer wavelets for rational dilations

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, also known as Lemarié–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 (1989 and 1992). 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 the one-dimensional result of Lemarié-Rieusset (1995).