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Borel’s theorem for $$C^{\infty}$$-functions on a non-Archimedean valued field. (English) Zbl 0579.26007
The following theorem has been proved here: Let K be a non-archimedean non-trivially valued field. Let $$\lambda_ 0,\lambda_ 1,..$$. be any sequence in K. Then there exists a $$C^{\infty}$$-function $$f: K\to K$$ such that $$D_ nf(0)=\lambda_ n$$ for all $$n\in \{0,1,...\}$$, where $$D_ n$$ is the operator of the n-th derivative defined as the continuous extension of the n-th difference quotient. This result was ”semi- published” before as Theorem 12.12 by the author [”Non-Archimedean calculus”, Report 7812. Nijmegen, The Netherlands: Mathematisch Instituut, Katholieke Universiteit (1978; Zbl 0463.26007)] and published without proof as Theorem 83.5 in the author’s book: ”Ultrametric calculus. An introduction to p-adic analysis” (1984; Zbl 0553.26006).
Reviewer: L.Márki

##### MSC:
 26E30 Non-Archimedean analysis 12J25 Non-Archimedean valued fields
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##### References:
 [1] D. Barsky : Fonctions k-lipschitziennes sur un anneau local et polynômes à valeurs entières . Bull. Soc. Math. Fr. 101, (1973) 397-411. · Zbl 0291.12107 · doi:10.24033/bsmf.1766 · numdam:BSMF_1973__101__397_0 · eudml:87216 [2] W.H. Schikhof : Non-archimedean calculus (Lecture notes) . Report 7812, Mathematisch Instituut, Katholieke Universiteit, Nijmegen (1978). · Zbl 0463.26007
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