Special \(p\)-adic analytic functions and Fourier transforms
type: Item
instance of: scholarly article
title: Special \(p\)-adic analytic functions and Fourier transforms
zbMATH Open document ID: 0860.11071
author: S. H. Smith
published in: Journal of Number Theory
publication date: 1997-04-24T00:00:00Z
review text: 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.
reviewed by: Wim Schikhof
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 46S10
Mathematics Subject Classification ID: 30G06
zbMATH DE Number: 955266
zbMATH Keywords: \(p\)-adic number field
zbMATH Keywords: \(p\)-adic analytic functions
zbMATH Keywords: antiderivative
zbMATH Keywords: special analytic functions
zbMATH Keywords: power series
zbMATH Keywords: Mahler series
MaRDI profile type: Publication
full work available at URL: jnth.1996.0130
OpenAlex ID: W1984807016
DOI: 10.1006/JNTH.1996.0130
description: scientific article; zbMATH DE number 955266