On the ultrametric Stone-Weierstrass theorem and Mahler's expansion

Paul-Jean Cahen; Jean-Luc Chabert

Journal de théorie des nombres de Bordeaux (2002)

  • Volume: 14, Issue: 1, page 43-57
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Cahen, Paul-Jean, and Chabert, Jean-Luc. "On the ultrametric Stone-Weierstrass theorem and Mahler's expansion." Journal de théorie des nombres de Bordeaux 14.1 (2002): 43-57. <http://eudml.org/doc/248912>.

@article{Cahen2002,
abstract = {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 $\hat\{E\}$ of $E$ in the completion $\hat\{V\}$ of $V$ is compact. We then show how to expand continuous functions in sums of polynomials.},
author = {Cahen, Paul-Jean, Chabert, Jean-Luc},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {ultrametric; Stone-Weierstrass theorems},
language = {eng},
number = {1},
pages = {43-57},
publisher = {Université Bordeaux I},
title = {On the ultrametric Stone-Weierstrass theorem and Mahler's expansion},
url = {http://eudml.org/doc/248912},
volume = {14},
year = {2002},
}

TY - JOUR
AU - Cahen, Paul-Jean
AU - Chabert, Jean-Luc
TI - On the ultrametric Stone-Weierstrass theorem and Mahler's expansion
JO - Journal de théorie des nombres de Bordeaux
PY - 2002
PB - Université Bordeaux I
VL - 14
IS - 1
SP - 43
EP - 57
AB - 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 $\hat{E}$ of $E$ in the completion $\hat{V}$ of $V$ is compact. We then show how to expand continuous functions in sums of polynomials.
LA - eng
KW - ultrametric; Stone-Weierstrass theorems
UR - http://eudml.org/doc/248912
ER -

References

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