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.


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


Zbl 0080.03504
Full Text: DOI Numdam EuDML


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