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On approximation by tight wavelet frames on the field of $p$-adic numbers. (arXiv:2307.06588v1 [math.NT])

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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


$\int_X|\lambda( {\chi})\hat{f}(\chi)|^2d\nu(\chi)<\infty$

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