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

TL;DR

This paper studies how well functions can be approximated using tight wavelet frames on the p-adic number field, providing bounds on approximation order for specific function classes.

Contribution

It introduces a new approximation framework using tight wavelet frames on $\,\mathbb{Q}_p$ and derives approximation order estimates for functions with certain spectral properties.

Findings

Derived approximation order bounds for functions with weighted spectral integrals.

Established conditions on the step function $\,\lambda(\chi)$ for effective approximation.

Extended wavelet approximation theory to the p-adic setting.

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 ${G}_n^\bot\setminus{G}_{n-1}^\bot$ 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 $\int_X|\lambda( {\chi})\hat{f}(\chi)|^2d\nu(\chi)<\infty$