Umbral calculus in non-archimedean analysis (Q2751745)

Continuing his previous investigations [Ann. Math. Blaise Pascal 5, No. 1, 55-73 (1998; Zbl 0907.46066)], the author develops, in the framework of \(p\)-adic analysis, some topics in the umbral calculus in the sense of \textit{G. C. Rota} [``Finite Operator Calculus'', New York (1975; Zbl 0328.05007)]. NEWLINENEWLINENEWLINELet \(p\) be a prime number, \(\mathbf Z_p\) the ring of \(p\)-adic integers, \(\mathbf Q_p \) the field of \(p\)-adic numbers, and let \(\mathbf K\) be a field with a complete valuation \(|\;|\) which contains \(\mathbf Q_p\). The paper is concerned with the study of continuous linear operators on the space \(C(\mathbf Z_p \to \mathbf Q_p) \) which commute with the translation operators. Here \(C(\mathbf Z_p \to \mathbf Q_p) \) denotes the space of continuous functions from \(\mathbf Z_p\) to \(\mathbf Q_p \) with the supremum norm. The results are applied to construct orthonormal bases for the space \(C(\mathbf Z_p \to \mathbf Q_p) \).NEWLINENEWLINEFor the entire collection see [Zbl 0969.00058].