Adelic versions of the Weierstrass approximation theorem

Abstract

Let $\underline{E}={\prod }_{p\in \mathbb{P}}{E}_p$ be a compact subset of $\hat{\mathbb{Z}}={\prod }_{p\in \mathbb{P}}{\mathbb{Z}}_p$ and denote by $\mathcal{C}(\underline{E},\hat{\mathbb{Z}})$ the ring of continuous functions from $\underline{E}$ into $\hat{\mathbb{Z}}$. We obtain two kinds of adelic versions of the Weierstrass approximation theorem. Firstly, we prove that the ring ${\mathrm{Int}}_{\mathbb{Q}}(\underline{E},\hat{\mathbb{Z}}):=\{f(x)\in \mathbb{Q}[x]\mid \forall p\in \mathbb{P},f(E_p)\subseteq {\mathbb{Z}}_p\}$ 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 $n\ge 0$, $\#(E_p(\mathrm{mod}p))>n$ for all but finitely many $p$, we prove the existence of regular bases of the $\mathbb{Z}$-module ${\mathrm{Int}}_{\mathbb{Q}}(\underline{E},\hat{\mathbb{Z}})$, and show that, for such a basis $\{f_n{\}}_{n\ge 0}$, every function $\underline{\phi }$ in ${\prod }_{p\in \mathbb{P}}\mathcal{C}(E_p,{\mathbb{Z}}_p)$ may be uniquely written as a series ${\sum }_{n\ge 0}{\underline{c}}_n{f}_n$ where ${\underline{c}}_n\in \hat{\mathbb{Z}}$ and ${\lim }_{n\to \infty }{\underline{c}}_n\to 0$.

minor corrections the statement of Theorem 3.5, which covers the case of a general compact subset of the profinite completion of Z. to appear in Journal of Pure and Applied Algebra, comments are welcome!

References in corpus (1)

• Polynomial overrings of $\mathrm{Int}(\mathbb{Z})$

Cited by in corpus (2)

• Polynomial Dedekind domains with finite residue fields of prime characteristic
• Nontriviality of rings of integral-valued polynomials