# Reprint: An interpolation series for continuous functions of a $p$-adic variable (1958)

### Kurt Mahler

## Abstract

Let $f(x)$ be a function on the set $I$ of $p$-adic integers. The subset $J$ of the non-negative integers is dense on $I$, hence a continuous function $f(x)$ on $I$ is already determined by its values on $J$, thus also by the numbers $a_{n}=∑_{k≥0}(−1)_{k}(kn )f(n−k)(n≥0).$ In this paper, Mahler proves that \( {a_n\} \) is a $p$-adic null sequence, and that $f(x)=∑_{n≥0}a_{n}(kn )$ for all $x∈I$. Thus, $f(x)$ can be approximated by polynomials. Mahler goes on to study conditions on the $a_{n}$ under which $f(x)$ is differentiable at a point or has a continuous derivative everywhere on $I$.

Reprint of the author's papers [J. Reine Angew. Math. 199, 23--34 (1958; Zbl 0080.03504); ibid. 208, 70--72 (1961; Zbl 0100.04003)].