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Non-archimedean umbral calculus. (English) Zbl 0907.46066

The famous K. Mahler’s theorem [J. Reine Angew. Math. 199, 23-34 (1958; Zbl 0080.03504)] states that every continuous function \(f: Z_p\to Q_p\) can be written as \(f(x)= \sum^\infty_{n= 0}a_n\left(\begin{smallmatrix} x\\ n\end{smallmatrix}\right)\), i.e. the functions \(\left(\begin{smallmatrix} x\\ n\end{smallmatrix}\right)\) form a basis of the Banach space \(C(Z_p\to Q_p)\). The present author, basing on his earlier results and on results of L. van Hamme gives constructions of different orthonormal basis for the space \(C(Z_p\to K)\), where \(K\) is a field extension of \(Q_p\), complete with respect to a non-Archimedean absolute value extending the \(p\)-adic absolute value.

MSC:

46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46E15 Banach spaces of continuous, differentiable or analytic functions
05A40 Umbral calculus
12J25 Non-Archimedean valued fields
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces

Citations:

Zbl 0080.03504
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References:

[1] Diarra, B., “Base de Mahler et autres”, Séminaire d’Analyse, 1994-95 (Aubière), exposé no. 16, 18pp., Sémin. Anal. Univ. Blaise Pascal (Clermont II), 10, Université Blaise Pascal (Clermont II), Clermont-Ferrand, 1997. · Zbl 0999.12014
[2] Mahler, K., “An Interpolation Series for Continuous Functions of a p-adic Variable”, Journal für reine und angewandte Mathematik, vol. 199, 1958, p. 23 - 34. · Zbl 0080.03504
[3] Roman, S., “The Umbral Calculus”, Academic Press, New York, 1984. · Zbl 0536.33001
[4] Rota, G.C., “Finite Operator Calculus”, Academic Press, New York, 1975. · Zbl 0328.05007
[5] Schikhof, W.H., “Ultrametric Calculus : An Introduction to p-adic Analysis”, Cambridge University Press, 1984. · Zbl 0553.26006
[6] Van Hamme, L., “Continuous Operators which commute with Translations, on the Space of Continuous Functions on Zp”, in “p-adic Functional Analysis”, BayodMartinez-Maurica[OK]De Grande - De Kimpe, p. 75-88, Marcel Dekker, 1992. · Zbl 0773.47039
[7] Verdoodt, A., “Normal Bases for Non-Archimedean Spaces of Continuous Functions”, Publicacions Matemàtiques, vol. 37, 1993, p. 403-427. · Zbl 0804.46088
[8] Verdoodt, A., “The Use of Operators for the Construction of Normal Bases for the Space of Continuous Functions on Vq”, Bulletin of the Belgian Mathematical Society - Simon Stevin, Vol. 1, 1994, p. 685-699. · Zbl 0814.46070
[9] A. Verdoodt, “Bases and Operators for the Space of Continuous Functions defined on a Subset of Zp”, thesis. · Zbl 0814.46070
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