Universal Taylor series have a strong form of universality
type: Item
instance of: scholarly article
title: Universal Taylor series have a strong form of universality
zbMATH Open document ID: 1156.30003
published in: Journal d'Analyse Mathématique
publication date: 2008-09-03T00:00:00Z
review text: The paper deals with the class \(U(\Omega,\xi)\) of holomorphic in a simply connected domain \(\Omega\) functions \(f\) whose Taylor series at \(\xi \in \Omega\) is universal with respect to overconvergence. It is considered the expansion of \(f\) in Faber series corresponding to any connected compact set \(\Gamma \subset \Omega\) with more than one point and such that \(\widehat{\mathbb{C}}\setminus \Gamma\) is connected (here \(\widehat{\mathbb{C}}\) denotes the Riemann sphere): \[ f(z)=\sum_{n=0}^{+\infty} c_n(f,\Gamma)F_n(\Gamma,z). \] Let \(S_n(f,\Gamma)(z)=\sum_{\nu=0}^{n} c_{\nu}(f,\Gamma)F_{\nu }(\Gamma,z)\) be a partial sum of the series. Definition 1. A holomorphic function \(f\in H(\Omega)\) belongs to the class \(U(\Omega,\Gamma)\) provided that for every compact set \(K\subset\mathbb{C}\) with \(K\cap\Omega=\emptyset\) and \(K^{\mathbb{C}}\) connected and every function \(h:K \to \mathbb{C}\) continuous on \(K\) and holomorphic in \(K^0\), there exists a sequence \((\lambda _n)\) of natural numbers such that \(S_{\lambda_n}(f,\Gamma)(z)\) converges to \(h(z)\) uniformly on \(K\) as \(n \to +\infty .\) Definition 2. A holomorphic function \(f\in H(\Omega)\) belongs to the class \(U_{\text{Fab}}(\Omega)\) provided that for every compact set \(K\subset\mathbb{C}\) with \(K\cap\Omega=\emptyset\) and \(K^{\mathbb{C}}\) connected and every function \(h:K \to \mathbb{C}\) continuous on \(K\) and holomorphic in \(K^0\), there exists a sequence \((\lambda _n)\) of natural numbers such that, for any connected compact set \(M\subset \Omega\) with more than one point and such that \(\widehat{\mathbb{C}}\setminus M\) is connected \[ \sup_{\Gamma \in M }\sup_{z \in K}| S_{\lambda_n}(f,\Gamma)(z)-h(z)| \to 0, \quad n \to +\infty. \] The authors obtain the following result, which answers a question of Mayenberger and Müller. Theorem. \(U(\Omega,\xi)=U(\Omega,\Gamma)=U_{\text{Fab}}(\Omega).\) The topology used here is the Carathéodory topology.
reviewed by: Olga M. Katkova
Mathematics Subject Classification ID: 30B30
Mathematics Subject Classification ID: 30B10
Mathematics Subject Classification ID: 30B40
zbMATH DE Number: 5320470
zbMATH Keywords: overconvergence
zbMATH Keywords: universal overconvergence
zbMATH Keywords: Faber series
author: Frédéric Bayart
author: V. Nestoridis
MaRDI profile type: Publication
full work available at URL: s11854-008-0017-5
OpenAlex ID: W2070216539
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DOI: 10.1007/S11854-008-0017-5
description: scientific article; zbMATH DE number 5320470