Abstract: This paper gives a review of multiresolution analysis and compactly supported orthogonal wavelets on Vilenkin groups. The Strang-Fix condition, the partition of unity property, the linear independence, the stability, and the orthonormality of "integer shifts" of the corresponding refinable functions are considered. Necessary and sufficient conditions are given for refinable functions to generate a multiresolution analysis in the L2 -spaces on Vilenkin groups. Several examples are provided to illustrate these results.
Keywords: Walsh transform, lacunary Walsh series, Cantor group, Vilenkin group, Vilenkin-Chrestenson transform, wavelets, stability.