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

Doc. Math. Extra Vol. Mahler Selecta, 599-614 (2019)
DOI: 10.25537/dm.2019.SB-599-614

Summary

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 $a_n=\sum_{k\ge 0} (-1)^k \binom{n}{k} f(n-k)\quad (n\ge 0).$ In this paper, Mahler proves that $\{a_n\}$ is a $p$-adic null sequence, and that $f(x)=\sum_{n\ge 0} a_n \binom{n}{k}$ 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 $I$. \par 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)].

Mathematics Subject Classification

11-03, 11S80, 26E30