Non-Archimedean Umbral Calculus
Copyright policy )Non-Archimedean Umbral Calculus
The famous \textit{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.
Non-Archimedean valued fields, Banach spaces of continuous, differentiable or analytic functions, Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis, Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, Umbral calculus, non-Archimedean absolute value, basis of the Banach space
Non-Archimedean valued fields, Banach spaces of continuous, differentiable or analytic functions, Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis, Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, Umbral calculus, non-Archimedean absolute value, basis of the Banach space
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