Step refinable functions and orthogonal MRA on $p$-adic Vilenkin groups

Abstract

We find the necessary and sufficient conditions for refinable step function under which this function generates an orthogonal MRA in the $L_2(\mathfrak{G})$ -spaces on Vilenkin groups $\mathfrak{G}$. We consider a class of refinable step functions for which the mask $m_0(\chi )$ is constant on cosets ${\mathfrak{G}}_{-1}^{\perp }$ and its modulus $|m_0(\chi )|$ takes two values only: 0 and 1. We will prove that any refinable step function $\phi$ from this class that generates an orthogonal MRA on $p$-adic Vilenkin group $\mathfrak{G}$ has Fourier transform with condition $\mathrm{supp}\hat{\phi }(\chi )\subset {\mathfrak{G}}_{p-2}^{\perp }$. We show the sharpness of this result too.

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