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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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