On approximation by tight wavelet frames on Vilenkin groups

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 $G_n^{\mathrm{\perp }}\setminus G_{n-1}^{\mathrm{\perp }}$ by equalities $\lambda (G_n^{\mathrm{\perp }}\setminus G_{n-1}^{\mathrm{\perp }})={\lambda }_n>0$ for which $\sum \frac{1}{{\lambda }_n}<\mathrm{\infty }$. We find the order of approximation of functions $f$ for which ${\int }_X\mathrm{\mid }\lambda (\chi )\widehat{f}(\chi ){\mathrm{\mid }}^{2}d\nu (\chi )<\mathrm{\infty }$. As a corollary, we obtain an approximation error for functions from Sobolev spaces with logarithmic weight.