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.