Fonctions analytiques et produits croulants. (Analytic functions and collapsing products)
type: Item
instance of: scholarly article
title: Fonctions analytiques et produits croulants. (Analytic functions and collapsing products)
zbMATH Open document ID: 0605.30046
author: Marie-Claude Sarmant
author: Alain Escassut
published in: Collectanea Mathematica
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/38385
review text: Let (K,\(| |)\) be a non-archimedean valued complete field that is algebraically closed. Let \(b_ 1,b_ 2,...\in K\) be such that \(\lim_{n\to \infty}| b_ n| =R\), \(| b_ n| <R\) for all n. Let \(\rho >0\) and set \(\Delta:=\{x\in K:| x| <R\), \(| x-b_ n| \geq \rho\) for all \(n\in {\mathbb{N}}\}\). Then there exists a Taylor series g such that \(g(b_ n)=0\) for all n, \(\lim_{| x| \uparrow R, x\in \Delta}| g(x)| =\infty\), and \(1+(1/g)\) is a collapsing product [M. C. Sarmant, Bull. Sci. Math., II. Sér. 109, 155- 178 (1985; Zbl 0564.12024)]. Further, if g is a Taylor series convergent for \(| x| <R\) and \(\lambda\in K\), then g can be factorized as \(\lambda ((1-\pi_{\lambda})/(1-\tau_{\lambda}))\), where \(\pi_{\lambda}\) and \(\tau_{\lambda}\) are collapsing products.
Mathematics Subject Classification ID: 30G06
Mathematics Subject Classification ID: 12J05
zbMATH DE Number: 3977390
zbMATH Keywords: collapsing product
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3977390