Coxeter Groups and Wavelet Sets

TL;DR

This paper explores the connection between traditional wavelet theory and fractal surface wavelets generated by reflections, establishing the existence of dilation-reflection wavelet sets for expansive matrices.

Contribution

It introduces the concept of dilation-reflection wavelet sets and proves their existence for arbitrary expansive matrix dilations, linking two wavelet theories.

Findings

Dilation-reflection wavelet sets exist for all expansive matrices.

Some sets serve as both dilation-translation and dilation-reflection wavelet sets.

The orthonormal structures in the two theories differ significantly.

Abstract

A traditional wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a system of unitary operators defined in terms of translation and dilation operations. A Coxeter/fractal-surface wavelet is obtained by defining fractal surfaces on foldable figures, which tesselate the embedding space by reflections in their bounding hyperplanes instead of by translations along a lattice. Although both theories look different at their onset, there exist connections and communalities which are exhibited in this semi-expository paper. In particular, there is a natural notion of a dilation-reflection wavelet set. We prove that dilation-reflection wavelet sets exist for arbitrary expansive matrix dilations, paralleling the traditional dilation-translation wavelet theory. There are certain measurable sets which can serve simultaneously as dilation-translation wavelet sets and dilation-reflection wavelet sets, although the orthonormal structures generated in the two theories are considerably different.