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 ${G}_n^\bot\setminus{G}_{n-1}^\bot$ by equalities $\lambda ({G}_n^\bot\setminus{G}_{n-1}^\bot)=\lambda_n>0$ for which $\sum\frac{1}{\lambda_n}<\infty$. We find the order of approximation of functions $f$ for which $\int_X|\lambda( {\chi})\hat{f}(\chi)|^2d\nu(\chi)<\infty$. As a corollary, we obtain an approximation error for functions from Sobolev spaces with logarithmic weight.