Orthogonal Wavelets on Direct Products of Cyclic Groups

We describe a method for constructing compactly supported orthogonal wavelets on a locally compact Abelian group $G$ which is the weak direct product of a countable set of cyclic groups of $p$th order. For all integers $p,n\ge 2$, we establish necessary and sufficient conditions under which the solutions of the corresponding scaling equations with $p^n$ numerical coefficients generate multiresolution analyses in $L^2(G)$. It is noted that the coefficients of these scaling equations can be calculated from the given values of $p^n$ parameters using the discrete Vilenkin–Chrestenson transform. Besides, we obtain conditions under which a compactly supported solution of the scaling equation in $L^2(G)$ is stable and has a linearly independent system of “integer” shifts. We present several examples illustrating these results.