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 $\lim_{ n\to \infty}\underline{ c}_n\to 0$.
Categories: math.NT math.RA