A wavelet basis for non-Archimedean $C^n$-functions and $n$-th Lipschitz functions

TL;DR

This paper constructs a wavelet basis for non-Archimedean function spaces, characterizing differentiability through basis coefficients and providing an orthonormal basis for $C^n(R,K)$ functions.

Contribution

It introduces a wavelet basis for $C(R,K)$ and characterizes $n$-times differentiable functions via this basis, offering a new analytical tool in non-Archimedean analysis.

Findings

Characterization of $n$-times differentiable functions via wavelet coefficients

Construction of an orthonormal basis for $C^n(R,K)$

Representation of continuous functions on non-Archimedean spaces

Abstract

A wavelet basis is a basis for the $K$-Banach space $C(R, K)$ of continuous functions from a complete discrete valuation ring $R$ whose residue field is finite to its quotient field $K$. In this paper, we prove a characterization of $n$-times continuously differentiable functions from $R$ to $K$ by the coefficients with respect to the wavelet basis and give an orthonormal basis for $K$-Banach space $C^n(R, K)$ of $n$-times continuously differentiable functions.