Isomorphism in Wavelets

Abstract

Two scaling functions ${\phi }_A$ and ${\phi }_B$ for Parseval frame wavelets are algebraically isomorphic, ${\phi }_A\simeq {\phi }_B$, 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 $d,s\ge 1$ respectively and let ${\phi }_A$ be a scaling function associated with matrix $A$ and generated by a finite solution. There always exists a scaling function ${\phi }_B$ associated with matrix $B$ such that \begin{equation*} φ_B \simeq φ_A. \end{equation*} An example shows that the assumption on the finiteness of the solutions can not be removed.