Equations For Parseval's Frame Wavelets In $L^2({\mathbb{R}}^d)$ With Compact Supports

Let $d\ge 1$ be a natural number and $A_0$ be a $d\times d$ expansive integral matrix with determinant $\pm 2.$ Then $A_0$ is integrally similar to an integral matrix $A$ with certain additional properties. A finite solution to the system of equations associated with the matrix $A$ will result in an iterated sequence $\{{\mathrm{\Psi }}^k{\chi }_{[0,1{)}^d}\}$ that converges to a function ${\phi }_A$ in $L^2({\mathbb{R}}^d)$-norm. With this (scaling) function ${\phi }_A,$ we will construct the Parseval’s wavelet function $\psi$ with compact support associated with matrix $A_0\mathrm{.}$