# Isomorphism in Wavelets

## TL;DR

This paper explores the algebraic isomorphism of scaling functions in Parseval frame wavelets, establishing conditions under which different matrices produce equivalent wavelet systems, with implications for their classification.

## Contribution

It demonstrates that for any given scaling function associated with a matrix, there exists an algebraically isomorphic scaling function linked to another matrix, highlighting the role of solution finiteness.

## Key findings

- Existence of algebraically isomorphic scaling functions for different matrices.
- Finiteness of solutions is essential for the isomorphism to hold.
- Counterexample shows the necessity of the finiteness assumption.

## Abstract

Two scaling functions $\varphi_A$ and $\varphi_B$ for Parseval frame wavelets are algebraically isomorphic, $\varphi_A \simeq \varphi_B$, if they have matching solutions to their (reduced) isomorphic systems of equations. Let $A$ and $B$ be $d\times d$ and $s\times s$ \thematrix matrices with $d, s\geq 1$ respectively and let $\varphi_A$ be a scaling function associated with matrix $A$ and generated by a finite solution. There always exists a scaling function $\varphi_B$ associated with matrix $B$ such that \begin{equation*} \varphi_B \simeq \varphi_A. \end{equation*} An example shows that the assumption on the finiteness of the solutions can not be removed.