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Continuous functions on compact subsets of local fields. (English) Zbl 0979.11054

J. Dieudonné [Bull. Sci. Math. (2) 68, 79–95 (1944; Zbl 0060.08204)] proved that every continuous function on a compact subset of \({\mathbb Q}_p\), the field of \(p\)-adic numbers, with values in \({\mathbb Q}_p\) can be uniformly approximated by polynomials, generalizing the classical Weierstrass approximation theorem. K. Mahler [J. Reine Angew. Math. 199, 23–34 (1958; Zbl 0080.03504); J. Reine Angew. Math. 208, 70–72 (1961; Zbl 0100.04003)] made explicit Dieudonné’s result for \(S={\mathbb Z}_p\), the ring of \(p\)-adic integers, showing that every continuous function \(f\) from \({\mathbb Z}_p\) to \({\mathbb Q} _p\) can be uniquely expressed in the form \[ f(x) = \sum _{n=0} ^{\infty} c_n \binom{x}{n} \] where \(c _n \to 0\) as \(n\to \infty\). Some years later, Y. Amice [Bull. Soc. Math. Fr. 92, 117–180 (1964; Zbl 0158.30203)] extended Mahler’s result to continuous functions on certain compact subsets \(S\) of a local field \(K\) taking values in \(K\).
In the paper under review, the authors generalize Amice’s theorem to any compact subset \(S\) of a local field \(K\). The construction given here uses a result of M. Bhargava [J. Reine Angew. Math. 490, 101–127 (1997; Zbl 0899.13022)] which associates to a given compact subset \(S\) of a local field \(K\) a sequence \(\{a_n\}_{n=0} ^{\infty}\subseteq S\), called the \(\pi\)-ordering \(\Lambda\), that minimizes the valuation of \(n!_\Lambda := (a_n - a_0) \cdots (a_n - a _{n-1})\). The main result is that given a continuous map \(f\) from \(S\) to \(K\), there exists a unique sequence \(\{c_n\}_{n=0}^{\infty}\) in \(K\) such that \[ f(x) = \sum_{n=0}^{\infty} c_n \binom{x}{n}_{\Lambda} \] for all \(x\in S\) and \(c_n \to 0\) as \(n\to\infty\). Here \(\displaystyle \binom{x}{n}_{\Lambda}:= {{(x-a_0) \cdots (x-a_{n-1})}\over{n!_{\Lambda}}}\). This result generalizes Mahler’s and Amice’s polynomials.

MSC:

11S05 Polynomials
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
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