Wavelet frames on Vilenkin groups and their approximation properties

TL;DR

This paper provides an explicit description of Walsh polynomial-generated tight wavelet frames on Vilenkin groups, introduces an algorithm for wavelet function construction, and studies their approximation properties, highlighting their potential for high approximation order.

Contribution

It offers a novel explicit characterization of Walsh polynomial-based tight wavelet frames on Vilenkin groups and an algorithm for constructing wavelet functions.

Findings

All wavelet functions can be compactly supported with arbitrarily high approximation order.

A general form for wavelet frames generated by Walsh polynomials is established.

Approximation properties of these frames are thoroughly analyzed.

Abstract

An explicit description of all Walsh polynomials generating tight wavelet frames is given. An algorithm for finding the corresponding wavelet functions is suggested, and a general form for all wavelet frames generated by an appropriate Walsh polynomial is described. Approximation properties of tight wavelet frames are also studied. In contrast to the real setting, it appeared that a wavelet tight frame decomposition has an arbitrary large approximation order whenever all wavelet functions are compactly supported.