On approximation by tight wavelet frames on the field of $p$-adic numbers

Abstract

We discuss the problem on approximation by tight step wavelet frames on the field ${\mathbb{Q}}_p$ of $p$-adic numbers. Let $G_n=\{x={\sum }_{k=n}^{\infty }{x}_k{p}^k\}$, $X$ 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 ${\int }_X|\lambda (\chi )\hat{f}(\chi ){|}^{2}d\nu (\chi )<\infty$

15 pages, 2 figures. arXiv admin note: text overlap with arXiv:2203.06352