# Wavelets on compact abelian groups

## TL;DR

This paper develops a multiresolution analysis framework on compact abelian groups using epimorphisms as dilation operators, characterizes scaling sequences, and constructs wavelet orthonormal bases in this setting.

## Contribution

It introduces a new characterization of scaling sequences for MRAs on compact abelian groups and constructs wavelet bases using these sequences.

## Key findings

- Characterization of scaling sequences for MRA on $L^p(G)$.
- Construction of wavelet orthonormal bases on $L^2(G)$.
- Extension of wavelet theory to compact abelian groups.

## Abstract

Multiresolution analysis (MRA) on compact abelian group $G$ has been constructed with epimorphism as a dilation operator. We show a characterization of scaling sequences of an MRA on $L^p(G)$, $1\le p<\infty$. With the help of this scaling sequence we construct a wavelet orthonormal basis of $L^2(G)$.