Orthogonal wavelets on direct products of cyclic groups
Yu. A. Farkov · Mathematical Notes · 2007
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 pth order. For all integers p, n ≥ 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.