Higher Rank Wavelets

TL;DR

This paper develops a higher rank multiresolution analysis framework enabling the construction of nonseparable wavelets, including Latin square and Meyer variants, with applications to orthonormal bases in multidimensional spaces.

Contribution

It introduces a new theory of higher rank MRAs and constructs novel nonseparable wavelets, expanding beyond tensor product approaches.

Findings

Constructed Latin square wavelets as rank 2 variants of Haar wavelets.

Developed nonseparable scaling functions for rank 2 Meyer wavelet variants.

Proved that compactly supported scaling functions for biscaled MRAs are necessarily separable.

Abstract

A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in $L^2(\bR^d)$. While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct Latin square wavelets as rank 2 variants of Haar wavelets. Also we construct nonseparable scaling functions for rank 2 variants of Meyer wavelet scaling functions, and we construct the associated nonseparable wavelets with compactly supported Fourier transforms. On the other hand we show that compactly supported scaling functions for biscaled MRAs are necessarily separable.