A characterization of wavelet sets on Vilenkin groups with its application to construction of MRA wavelets

TL;DR

This paper characterizes wavelet sets on Vilenkin groups, offering a new approach to constructing wavelets and exploring their relation to multiresolution analyses, thereby advancing wavelet theory on these groups.

Contribution

It provides a new characterization of wavelet sets on Vilenkin groups, enabling alternative wavelet construction methods and clarifying their connection to multiresolution analyses.

Findings

Established a characterization of wavelet sets on Vilenkin groups.

Provided a new method for constructing wavelets from wavelet sets.

Linked wavelet sets to multiresolution analyses, answering prior open questions.

Abstract

Let $G$ be a Vilenkin group. In 2008, Y. A. Farkov constructed wavelets on $G$ via the multiresolution analysis method. In this article, a characterization of wavelet sets on $G$ is established, which provides another method for the construction of wavelets. As an application, the relation between multiresolution analyses and wavelets determined from wavelet sets is also presented. To some extent, these results positively answer a question mentioned by P. Mahapatra and D. Singh in [Bull. Sci. Math. 167 (2021), Paper No. 102945, 20 pp].