Representation theory and multilevel filters

TL;DR

This paper develops a general framework for wavelet filters and multiresolution analysis using iterated function system measures, extending classical theories to more complex fractal and Julia set cases.

Contribution

It introduces a new setting for wavelet filters based on IFS measures, with filters indexed by an infinite-dimensional loop group, broadening the scope of multilevel filter theory.

Findings

Framework includes classical and Julia set cases

Wavelet filters indexed by an infinite-dimensional loop group

Extends multiresolution analysis beyond traditional L^2 spaces

Abstract

We present a general setting where wavelet filters and multiresolution decompositions can be defined, beyond the classical $\mathbf L^2(\mathbb R,dx)$ setting. This is done in a framework of {\em iterated function system} (IFS) measures; these include all cases studied so far, and in particular the Julia set/measure cases. Every IFS has a fixed order, say $N$, and we show that the wavelet filters are indexed by the infinite dimensional group $G$ of functions from $X$ into the unitary group $U_N$. We call $G$ the loop group because of the special case of the unit circle.