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

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.