On the ultrametric Stone-Weierstrass theorem and Mahler's expansion
Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 43-57.

Nous explicitons une version ultramétrique du théorème de Stone-Weierstrass. Pour une partie E d’un anneau de valuation V de hauteur 1, nous montrons, sans aucune hypothèse sur le corps résiduel, que l’ensemble des fonctions polynomiales est dense dans l’anneau des fonctions continues de E dans V si et seulement si la clôture topologique E^ de E dans le complété de V^ est compacte. Nous explicitons ainsi le développement d’une fonction continue en série de fonctions polynomiales.

We describe an ultrametric version of the Stone-Weierstrass theorem, without any assumption on the residue field. If E is a subset of a rank-one valuation domain V, we show that the ring of polynomial functions is dense in the ring of continuous functions from E to V if and only if the topological closure E^ of E in the completion V^ of V is compact. We then show how to expand continuous functions in sums of polynomials.

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     title = {On the ultrametric {Stone-Weierstrass} theorem and {Mahler's} expansion},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     publisher = {Universit\'e Bordeaux I},
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Cahen, Paul-Jean; Chabert, Jean-Luc. On the ultrametric Stone-Weierstrass theorem and Mahler's expansion. Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 43-57. https://www.numdam.org/item/JTNB_2002__14_1_43_0/

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