Adelic versions of the Weierstrass approximation theorem

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.