A method for constructing orthonormal bases for non-Archimedean Banach spaces of continuous functions
Let \(K\) be a locally compact non-trivially non-Archimedean valued field. Let \(\alpha\in K\), \(|\alpha|= \max\{|x|: x\in K, |x|< 1\}\). A subset \(M\) of the unit ball of \(K\) is called regular compact if \(M\) is compact and there exist numbers \(q_1,q_2,\dots\) such that each ball in \(M\) of diameter \(|\alpha|^n\) decomposes into \(q_{n+1}\) balls of diameter \(|\alpha|^{n+1}\). Will the help of so-called very well distributed sequences, based on the theory of \textit{Y. Amice} [Bull. Soc. Math. Fr. 92, 117--180 (1964; Zbl 0158.30203)] various methods are given to construct orthogonal bases of \(C(M\to K)\). As a corollary it is shown, for example that \((f_n)\), and \((f^n_n)\) are orthonormal bases of \(C(\mathbb{Z}_p\to \mathbb{Q}_p)\) where \(f_n= \sum^\infty_{i=0} p^i\left(\begin{smallmatrix} x\\ i+n\end{smallmatrix}\right)\). Many more examples are given.
scientific article; zbMATH DE number 1501765