Adelic versions of the Weierstrass approximation theorem

Abstract: Let $\underline{E}=\prod_{p\in\mathbb{P}}E_p$ be a compact subset of $\widehat{\mathbb{Z}}=\prod_{p\in\mathbb{P}}\mathbb{Z}p$ and denote by $\mathcal C(\underline{E},\widehat{\mathbb{Z}})$ the ring of continuous functions from $\underline{E}$ into $\widehat{\mathbb{Z}}$. We obtain two kinds of adelic versions of the Weierstrass approximation theorem. Firstly, we prove that the ring ${\rm Int}{\mathbb{Q}}(\underline{E},\widehat{\mathbb{Z}}):={f(x)\in\mathbb{Q}[x]\mid \forall p\in\mathbb{P},\;\;f(E_p)\subseteq \mathbb{Z}p}$ is dense in the direct product $\prod{p\in\mathbb{P}}\mathcal C(E_p,\mathbb{Z}p)\,$ for the uniform convergence topology. Secondly, under the hypothesis that, for each $n\geq 0$, $#(E_p\pmod{p})>n$ for all but finitely many $p$, we prove the existence of regular bases of the $\mathbb{Z}$-module ${\rm Int}{\mathbb{Q}}(\underline{E},\widehat{\mathbb{Z}})$, and show that, for such a basis ${f_n}{n\geq 0}$, every function $\underline{\varphi}$ in $\prod{p\in\mathbb{P}}\mathcal{C}(E_p,\mathbb{Z}p)$ may be uniquely written as a series $\sum{n\geq 0}\underline{c}n f_n$ where $\underline{c}_n\in\widehat{\mathbb{Z}}$ and $\lim{n\to \infty}\underline{c}_n\to 0$.