Special \(p\)-adic analytic functions and Fourier transforms
For an analytic function \(z \mapsto \sum a_nz^n\) with domain \(\mathbb{Z}_p\) and range \(\mathbb{C}_p\) (the completion of the algebraic closure of the \(p\)-adic number field), its natural antiderivative \(z\mapsto \sum a_n z^{n+1}/(n+1)\) may not be defined on the whole of \(\mathbb{Z}_p\). To overcome this difficulty the author considers the subclass of all analytic functions \(z\mapsto \sum a_nz^n\) for which \(\lim_n n!\pi^{-n} a_n=0\) (where \(\pi\in \mathbb{C}_p\) is fixed, \(|\pi |= p^{-{1 \over p-1}})\), together to form a Banach algebra SA (with respect to the norm \(\sup_n |n! \pi^{-n} a_n |)\) of the so-called special analytic functions. Besides its invariance for (anti)derivatives the space SA turns out to have quite some additional, remarkable, invariance properties, such as for affine changes of variable, for indefinite sums (the inverse of the difference operator), under the natural convolutions \(*\) and \(\otimes\), introduced by the author in previous papers. Characterizations of SA in terms of expansions as ordinary, \(*\)-, \(\otimes\)-power series, and Mahler series are given. This lucid paper is a must to those who are interested in the elementary theory of \(p\)-adic functions.
scientific article; zbMATH DE number 955266