## 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

Zbl 0080.03504
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### References:

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