  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.
