We describe an ultrametric version of the Stone-Weierstrass theorem, without any assumption on the residue field. If is a subset of a rank-one valuation domain , we show that the ring of polynomial functions is dense in the ring of continuous functions from to if and only if the topological closure of in the completion of is compact. We then show how to expand continuous functions in sums of polynomials.
Nous explicitons une version ultramétrique du théorème de Stone-Weierstrass. Pour une partie d’un anneau de valuation de hauteur , nous montrons, sans aucune hypothèse sur le corps résiduel, que l’ensemble des fonctions polynomiales est dense dans l’anneau des fonctions continues de dans si et seulement si la clôture topologique de dans le complété de est compacte. Nous explicitons ainsi le développement d’une fonction continue en série de fonctions polynomiales.
@article{JTNB_2002__14_1_43_0, author = {Paul-Jean Cahen and Jean-Luc Chabert}, title = {On the ultrametric {Stone-Weierstrass} theorem and {Mahler's} expansion}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {43--57}, publisher = {Universit\'e Bordeaux I}, volume = {14}, number = {1}, year = {2002}, zbl = {1031.46086}, mrnumber = {1925989}, language = {en}, url = {https://jtnb.centre-mersenne.org/item/JTNB_2002__14_1_43_0/} }
TY - JOUR AU - Paul-Jean Cahen AU - Jean-Luc Chabert TI - On the ultrametric Stone-Weierstrass theorem and Mahler's expansion JO - Journal de théorie des nombres de Bordeaux PY - 2002 SP - 43 EP - 57 VL - 14 IS - 1 PB - Université Bordeaux I UR - https://jtnb.centre-mersenne.org/item/JTNB_2002__14_1_43_0/ LA - en ID - JTNB_2002__14_1_43_0 ER -
%0 Journal Article %A Paul-Jean Cahen %A Jean-Luc Chabert %T On the ultrametric Stone-Weierstrass theorem and Mahler's expansion %J Journal de théorie des nombres de Bordeaux %D 2002 %P 43-57 %V 14 %N 1 %I Université Bordeaux I %U https://jtnb.centre-mersenne.org/item/JTNB_2002__14_1_43_0/ %G en %F JTNB_2002__14_1_43_0
Paul-Jean Cahen; Jean-Luc Chabert. On the ultrametric Stone-Weierstrass theorem and Mahler's expansion. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 1, pp. 43-57. https://jtnb.centre-mersenne.org/item/JTNB_2002__14_1_43_0/
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