Step Refinable Functions and Orthogonal MRA on Vilenkin Groups

Abstract

We find necessary and sufficient conditions on refinable step function under which this function generates an orthogonal MRA in the \(L_{2}(\mathfrak{G})\)-spaces on Vilenkin group \(\mathfrak{G}\). We consider a class of refinable step functions for which the mask m0(χ) is constant on cosets \(\mathfrak{G}_{-1}^{\bot}\chi\) and its modulus |m0(χ)| has two values only: 0 and 1. We prove that any refinable step function φ from this class that generates an orthogonal MRA on Vilenkin group \(\mathfrak{G}\) has Fourier transform with condition \(\operatorname{supp}\hat{\varphi}(\chi)\subset\mathfrak{G}_{p-2}^{\bot}\). We show the sharpness of this result, too.