Universal Taylor series on doubly connected domains
type: Item
instance of: scholarly article
title: Universal Taylor series on doubly connected domains
zbMATH Open document ID: 1178.30002
author: Athanassia G. Bacharoglou
published in: Results in Mathematics
publication date: 2009-06-17T00:00:00Z
review text: Let \(K\subset\mathbb{C}\) be a connected compact set such that \(\Omega=\mathbb{C}\setminus K\) is connected, and \(\zeta_0\in\Omega\). Also, let \(U^\infty(\Omega,\zeta_0)\) denote the class of holomorphic functions \(f:\Omega\to\mathbb{C}\) said to be `universal' with respect to \(\Omega\) and \(\zeta_0\) if the following holds: for every polynomial \(h\), there exists a sequence \(\{\lambda_n\}\) of non-negative integers such that, for all \[ l=0,1,2,\dots,\;\lim_{n\to\infty}\left(\sup_{z\in K}\left| \frac{\partial^l}{\partial z^l}S_{\lambda_n}(f,\zeta_0)(z)-h^{(l)}(z) \right|\right)=0. \] The existence of universal Taylor series was established by \textit{V. Nestoridis} [Ann. Inst. Fourier 46, No. 5, 1293--1306 (1996; Zbl 0865.30001)]. The author proves that, for every \(\zeta_0\in\Omega\), the class \(U^\infty(\Omega,\zeta_0)\) is \(G_\delta\) and dense in \(H(\Omega)\) and so non-empty; and \(U^\infty(\Omega, \zeta_0)\{0\}\) contains a dense vector subspace of \(H(\Omega)\). (This extends an earlier result of \textit{A. Melas} [Ann. Inst. Fourier 51, 1539--1551 (2001; Zbl 0989.30003)]). He also proves the same results in the subspace \(H_0(\Omega)\) of \(H(\Omega)\) which contains those functions \(f\in H(\Omega)\) for which \(\lim_{z\to\infty}(z)=0\).
Mathematics Subject Classification ID: 30B30
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 5565946
zbMATH Keywords: Taylor series
zbMATH Keywords: universal function
zbMATH Keywords: Baire's theorem
zbMATH Keywords: Runge's approximation theorem
zbMATH Keywords: algebraic genericity
reviewed by: David A. Brannan
MaRDI profile type: Publication
full work available at URL: s00025-008-0325-1
OpenAlex ID: W1990442656
DOI: 10.1007/S00025-008-0325-1
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description: scientific article; zbMATH DE number 5565946