  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.
