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

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 $a_n={\sum }_{k\ge 0}(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k})f(n-k)(n\ge 0).$ In this paper, Mahler proves that \( {a_n\} \) is a $p$-adic null sequence, and that $f(x)={\sum }_{n\ge 0}{a}_n\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ for all $x\in I$. 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)].