The unitary extension principle on locally compact abelian groups

TL;DR

This paper generalizes the unitary extension principle to locally compact abelian groups, enabling the construction of wavelet frames using modulations and shift-invariant systems, with explicit examples on various groups.

Contribution

It extends the UEP framework from Euclidean spaces to locally compact abelian groups, broadening the scope of wavelet frame construction methods.

Findings

Generalized UEP applicable to locally compact abelian groups

Constructed frames using B-splines and characteristic functions

Provided explicit examples on specific groups

Abstract

The unitary extension principle (UEP) by Ron and Shen yields conditions for the construction of a multi-generated tight wavelet frame for $L^2(\mr^s)$ based on a given refinable function. In this paper we show that the UEP can be generalized to locally compact abelian groups. In the general setting, the resulting frames are generated by modulates of a collection of functions, via the Fourier transform this corresponds to a generalized shift-invariant system. Both the stationary and the nonstationary case are covered. We provide general constructions, based on B-splines on the group itself as well as on characteristic functions on the dual group. Finally, we consider a number of concrete groups and derive explicit constructions of the resulting frames.