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The Weierstrass-Stone approximation theorem for \(p\)-adic \(C^ n\)- functions. (English) Zbl 0821.26019

Authors’ abstract: “Let \(K\) be a non-Archimedean valued field. Then, on compact subsets of \(K\), every \(K\)-valued \(C^ n\)-function can be approximated in the \(C^ n\)-topology by polynomial functions. This result is extended to a Weierstrass-Stone type theorem.”
Note that, as is well explained in the introduction, the standard proof for the classical Archimedean case does not work here, lacking an indefinite integral and the mean value theorem.

MSC:

26E30 Non-Archimedean analysis
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References:

[1] van Rooij, A.C.M.Non-Archimedean Functional Analysis. Marcel Dekker, New York, 1978. · Zbl 0396.46061
[2] Schikhof, W.H.: Non-Archimedean Calculus. Report 7812, Mathematisch Instituut, Katholieke Universiteit, Nijmegen, The Netherlands, 1978. · Zbl 0463.26007
[3] Schikhof, W.H.: Ultrametric Calculus. Cambridge University Press, 1982. · Zbl 0553.26006
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