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arXiv · 1211.2633

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

S. F. Lukomskii
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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 $m_0(χ)$ is constant on cosets $\mathfrak G_{-1}^\bot$ and its modulus $|m_0(χ)|$ takes two values only: 0 and 1. We will prove that any refinable step function $φ$ from this class that generates an orthogonal MRA on $p$-adic Vilenkin group $\mathfrak G$ has Fourier transform with condition ${\rm supp} \hat φ(χ)\subset \mathfrak G_{p-2}^\bot$. We show the sharpness of this result too.

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S. F. Lukomskii. 2012-11-12. Step refinable functions and orthogonal MRA on $p$-adic Vilenkin groups. https://arxiv.org/abs/1211.2633

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