Orthogonal and Periodic Wavelets on Vilenkin Groups

Abstract

As noted in Chap. 1, the Walsh function can be identified with characters of the Cantor dyadic group. This fact was first recognized by Gelfand in the 1940s, who offered to Vilenkin study series with respect to characters of a large class of abelian groups which includes the Cantor group as special case see Vilenkin [1], Fine [2], Agaev, Vilenkin, Dzhafarli, Rubinshtein [3]. For wavelets on Vilenkin groups most of the results relate to the locally compact group $G_{p}$, which is defined by a fixed integer $p\ge 2.$ The group $G_{p}$ has a standard interpretation on $\mathbb {R_{+}}$. Since the case $p=2$ corresponds to the Cantor group $\mathscr {C}$, all the results on wavelets on $\mathbb {R_{+}}$ presented in Chap. 4 can be rewritten for wavelets on $\mathscr {C}$. In this section, necessary and sufficient conditions are given for refinable functions to generate an MRA in the space $L^{2}(G_{p})$. The partition of unity property, the linear independence, the stability, and the orthogonality of “integer shifts” of refinable functions in $L^{2}(G_{p})$ are also considered.