Adelic versions of the Weierstrass approximation theorem

Abstract

Let E_=∏p∈PEp be a compact subset of Zˆ=∏p∈PZp and denote by C(E_,Zˆ) the ring of continuous functions from E_ into Zˆ. We obtain two kinds of adelic versions of the Weierstrass approximation theorem. Firstly, we prove that the ring IntQ(E_,Zˆ):={f(x)∈Q[x]|f(E_)⊆Zˆ} is dense in the product ∏p∈PC(Ep,Zp) for the uniform convergence topology. We also obtain an analogous statement for general compact subsets of Zˆ.Secondly, under the hypothesis that, for each n≥0, #(Ep(modp))>n for all but finitely many primes p, we prove the existence of regular bases of the Z-module IntQ(E_,Zˆ), and show that, for such a basis {fn}n≥0, every function φ_ in ∏p∈PC(Ep,Zp) may be uniquely written as a series ∑n≥0c_nfn where c_n∈Zˆ and limn→∞⁡c_n→0. Moreover, we characterize the compact subsets E_ for which the ring IntQ(E_,Zˆ) admits a regular basis in Pólya's sense by means of an adelic notion of ordering which generalizes Bhargava's p-ordering.