Orthonormal dilations of Parseval wavelets

TL;DR

This paper demonstrates that any Parseval wavelet frame can be viewed as a projection of an orthonormal wavelet basis linked to the Baumslag-Solitar group, providing explicit examples and analyzing their structural properties.

Contribution

It establishes a method to obtain orthonormal dilations of Parseval wavelets via group representations and explores their structure through symbolic dynamics.

Findings

Parseval wavelet frames are projections of orthonormal bases for Baumslag-Solitar group representations

Explicit examples of orthonormal dilations of Parseval wavelets are provided

Some Parseval wavelet sets admit infinitely many non-isomorphic orthonormal dilations

Abstract

We prove that any Parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the Baumslag-Solitar group $$BS(1,2)=< u,t | utu^{-1}=t^2>.$$ We give a precise description of this representation in some special cases, and show that for wavelet sets, it is related to symbolic dynamics. We show that the structure of the representation depends on the analysis of certain finite orbits for the associated symbolic dynamics. We give concrete examples of Parseval wavelets for which we compute the orthonormal dilations in detail; we show that there are examples of Parseval wavelet sets which have infinitely many non-isomorphic orthonormal dilations.