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

  • Kurt Mahler


Let f(x)f(x) be a function on the set II of pp-adic integers. The subset JJ of the non-negative integers is dense on II, hence a continuous function f(x)f(x) on II is already determined by its values on JJ, thus also by the numbers an=k0(1)k(nk)f(nk)(n0).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 pp-adic null sequence, and that f(x)=n0an(nk)f(x)=\sum_{n\ge 0} a_n \binom{n}{k} for all xIx\in I. Thus, f(x)f(x) can be approximated by polynomials. Mahler goes on to study conditions on the ana_n under which f(x)f(x) is differentiable at a point or has a continuous derivative everywhere on II.

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)].