Schikhof, W. H. Borel’s theorem for \(C^{\infty}\)-functions on a non-Archimedean valued field. (English) Zbl 0579.26007 Compos. Math. 55, 289-294 (1985). 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 Cited in 1 Document MSC: 26E30 Non-Archimedean analysis 12J25 Non-Archimedean valued fields Keywords:Borel’s theorem; non-Archimedean calculus; non-archimedean non-trivially valued field; \(C^{\infty }\)-function Citations:Zbl 0463.26007; Zbl 0553.26006 PDFBibTeX XMLCite \textit{W. H. Schikhof}, Compos. Math. 55, 289--294 (1985; Zbl 0579.26007) Full Text: Numdam EuDML 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 [2] W.H. Schikhof : Non-archimedean calculus (Lecture notes) . Report 7812, Mathematisch Instituut, Katholieke Universiteit, Nijmegen (1978). · Zbl 0463.26007 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.