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The van der Put base for \(C^ n\)-functions. (English) Zbl 0827.46067

Let \(K\) be an algebraic extension of \(Q_p\), the field of \(p\)-adic numbers, and \(Z_p\) the ring of \(p\)-adic integers. Write \(\varphi_1 f(x_1, x_2)= (f(x_2)- f(x_1))/ (x_2- x_1)\) and \(\varphi_2 f(x_1, x_2, x_3)= (\varphi_1 f(x_2, x_3)- \varphi_1 f(x_1, x_3))/ (x_2- x_1)\) where \(x_1\), \(x_2\), \(x_3\) are all distinct. Then \(C^2 (Z_p\to K)\) denotes the Banach space of \(f: Z_p\to K\) such that \(\varphi_2 f\) can be extended to a continuous function on \(Z^3_p\). The author constructed an orthonormal base for \(C^2 (Z_p\to K)\) and claimed that the same holds for \(C^n (Z_p\to K)\). The proof is highly computational. For reference, see W. H. Schikhof [Ultrametric calculus: An introduction to \(p\)-adic analysis (1984; Zbl 0553.26006)].

MSC:

46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
26E30 Non-Archimedean analysis

Citations:

Zbl 0553.26006
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