Adelic versions of the Weierstrass approximation theorem

Let $\uE=prod_{pinSP}E_p$ be a compact subset of $hZ=prod_{pinSP}SZ_p$ and denote by $Cc(\uE,hZ)$ the ring of continuous functions from $\uE$ into $hZ$. We obtain two kinds of adelic versions of the Weierstrass approximation theorem. Firstly, we prove that the ring $int_SQ(\uE,hZ):={f(x)inSQ[x]mid f(\uE)subseteq hZ}$ is dense in the product $prod_{pinSP}Cc(E_p,SZ_p),$ for the uniform convergence topology. We also obtain an analogous statement for general compact subsets of $hZ$. Secondly, under the hypothesis that, for each $ngeq 0$, $#(E_ppmod{p})>n$ for all but finitely many primes $p$, we prove the existence of regular bases of the $SZ$-module $int_SQ(\underline{E},hZ)$, and show that, for such a basis ${f_n}_{ngeq 0}$, every function $\ufi$ in $prod_{pinSP}Cc(E_p,SZ_p)$ may be uniquely written as a series $sum_{ngeq 0}\uc_n f_n$ where $\uc_ninhZ$ and $lim_{n o infty}\uc_n o 0$. Moreover, we characterize the compact subsets $\uE$ for which the ring $int_SQ(\uE,hZ)$ admits a regular basis in P'olya's sense by means of an adelic notion of ordering which generalizes Bhargava's $p$-ordering.