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Adelic versions of the Weierstrass approximation theorem

Abstract : Let (E) under bar = Pi(p is an element of P) E-p be a compact subset of (Z) over cap = Pi(p is an element of P) Z(p) and denote by C((E) over bar, (Z) over cap the ring of continuous functions from (E) over bar into (Z) over cap. We obtain two kinds of adelic versions of the Weierstrass approximation theorem. Firstly, we prove that the ring Int(Q)((E) over bar, (Z) over cap) := \f (x) is an element of Q[x] vertical bar f((E) over bar) subset of (Z) over cap\ is dense in the product Pi(p is an element of P) C(E-p, Z(p)) for the uniform convergence topology. We also obtain an analogous statement for general compact subsets of (Z) over cap. Secondly, under the hypothesis that, for each n >= 0, #(E-p (mod p)) > n for all but finitely many primes p, we prove the existence of regular bases of the Z-module Int(Q)((E) under bar, (Z) over cap), and show that, for such a basis \f(n)\(n >= 0), every function (phi) under bar in Pi(p is an element of P) C(E-p, Z(p)) may be uniquely written as a series Sigma(n >= 0)(c) under bar (n)f(n) where (c) under barn is an element of (Z) over cap and lim(n ->infinity) (c) under bar (n) -> 0. Moreover, we characterize the compact subsets (E) under bar for which the ring Int(Q)((E) under bar, (Z) over cap) admits a regular basis in Polya's sense by means of an adelic notion of ordering which generalizes Bhargava's p-ordering. (C) 2017 Elsevier B.V. All rights reserved.
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Soumis le : lundi 28 mars 2022 - 09:37:48
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Jean-Luc Chabert, Giulio Peruginelli. Adelic versions of the Weierstrass approximation theorem. Journal of Pure and Applied Algebra, Elsevier, 2018, 222 (3), pp.568-584. ⟨10.1016/j.jpaa.2017.04.020⟩. ⟨hal-03621213⟩

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