Wavelet filters and infinite-dimensional unitary groups

TL;DR

This paper explores the harmonic analysis of wavelet filters, focusing on their dependence on scale and genus, and utilizes infinite-dimensional unitary groups to study filter extension problems.

Contribution

It introduces a novel analysis of wavelet filters using representations of the C^*-algebra O_N and the infinite-dimensional group of unitary maps, addressing filter extension issues.

Findings

Wavelet filters' harmonic analysis is based on C^*-algebra O_N representations.

The infinite-dimensional group of unitary maps U(N) plays a key role in the analysis.

Extension problems from low-pass to multiresolution filters are studied using this group.

Abstract

In this paper, we study wavelet filters and their dependence on two numbers, the scale N and the genus g. We show that the wavelet filters, in the quadrature mirror case, have a harmonic analysis which is based on representations of the C^*-algebra O_N. A main tool in our analysis is the infinite-dimensional group of all maps T -> U(N) (where U(N) is the group of all unitary N-by-N matrices), and we study the extension problem from low-pass filter to multiresolution filter using this group.