On approximation by tight wavelet frames on Vilenkin groups

Abstract

We consider the approximate properties of tight wavelet frames on Vilenkin group $G$. Let $\{G_n{\}}_{n\in \mathbb{Z}}$ be a main chain of subgroups, $X$ be a set of characters. We define a step function $\lambda (\chi )$ that is constant on cosets $G_n^{\perp }\setminus G_{n-1}^{\perp }$ by equalities $\lambda (G_n^{\perp }\setminus G_{n-1}^{\perp })={\lambda }_n>0$ for which $\sum \frac{1}{{\lambda }_n}<\infty$. We find the order of approximation of functions $f$ for which ${\int }_X|\lambda (\chi )\hat{f}(\chi ){|}^{2}d\nu (\chi )<\infty$. As a corollary, we obtain an approximation error for functions from Sobolev spaces with logarithmic weight.

arXiv comment18 pages, 3 figures. arXiv admin note: text overlap with arXiv:2203.06352, arXiv:2307.06588