Continuous functions on compact subsets of local fields

where the sequence cn tends to 0 as n→∞. The purpose of this paper is to extend Mahler’s theorem to continuous functions from any compact subset S of a local field K to K. Here by a local field we mean the fraction field of a complete discrete valuation ring R whose residue field k = R/πR is finite. Our theorem implies, in particular, that every continuous function from S to K can be uniformly approximated by polynomials. This generalization of Weierstrass’s approximation theorem was first proved in the case K = Qp by Dieudonné [3]. Mahler [4] made explicit Dieudonné’s result in the case S = Zp by giving a canonical polynomial interpolation series for the continuous functions from Zp to Qp. Amice [1] later extended Mahler’s theorem to continuous functions on certain “very well-distributed” subsets S of a local field K. The present work provides canonical polynomial interpolation series for all S and K, and thus constitutes a best possible generalization of Mahler’s result in this context. The main ingredient in our work is a generalization of the binomial polynomials ( x n ) introduced by the first author [2]. Their construction is as follows. Given a subset S ⊂ K, fix a π-ordering Λ of S, which is a sequence a0, a1, . . . in which an ∈ S is chosen to minimize the valuation of (an−a0) · · · (an−an−1). It is a fundamental lemma [2, Theorem 1] that the generalized factorial n!Λ = (an − a0) · · · (an − an−1)