Mahler, Kurt

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


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