N-valid trees in wavelet theory on Vilenkin groups

TL;DR

This paper characterizes when certain step functions generate multiresolution analyses on p-adic Vilenkin groups, linking this property to special N-valid rooted trees, thus advancing wavelet theory in non-Archimedean settings.

Contribution

It establishes a novel criterion connecting N-valid rooted trees with the generation of MRAs by elementary step functions on p-adic Vilenkin groups.

Findings

N-valid rooted trees characterize MRA generation

Step functions generate MRAs iff associated with N-valid trees

Provides a combinatorial approach to wavelet construction on Vilenkin groups

Abstract

We consider a class of $(N,M)$-elementary step functions on the $p$-adic Vilenkin group. We prove that $(N,M)$-elementary step function generates a MRA on $p$-adic Vilenkin group iff it is generated by a special $N$-valid rooted tree on the set of vertices $\{0,1,\dots p-1\}$ with the vector $(0,...,0)\in \mathbb Z^N$ as a root. Bibliography: 15 titles.