Boundedness, regularity and smoothness of universal Taylor series
type: Item
instance of: scholarly article
title: Boundedness, regularity and smoothness of universal Taylor series
zbMATH Open document ID: 1112.30003
published in: Archiv der Mathematik
publication date: 2006-12-06T00:00:00Z
review text: The author improves results of \textit{G.~Costakis} [Math. Proc. Camb. Philos. Soc. 128, 157--175 (2000; Zbl 0956.30003)] on the existence of universal holomorphic functions. To be more precise, let \(\Omega \subset \mathbb C\) be an unbounded simply connected domain such that \((\overline{\Omega})^\circ = \Omega\), \((\mathbb C \setminus \overline{\Omega}) \cup \{\infty\}\) is connected and locally connected at \(\infty\). For example, \(\Omega\) might be an open half plane. Furthermore, let \(A(\Omega)\) denote the space of all functions which are continuous on \(\overline{\Omega}\) and holomorphic in \(\Omega\), and let \(H^\infty(\Omega)\) denote the space of all functions which are holomorphic and bounded in \(\Omega\). Finally, let \(X = A(\Omega) \cap H^\infty(\Omega)\). Then \(X\) equipped with the supremum norm on \(\overline{\Omega}\) is a Banach space. For \(f \in X\) and \(\zeta \in \Omega\), let \[ S_n(f,\zeta)(z) = \sum_{k=0}^n \frac{f^{(k)}(\zeta)}{k!}\,(z-\zeta)^k \] be the \(n\)-th partial sum of the Taylor series of \(f\) around the point \(\zeta\). A function \(f \in X\) is called universal in the weak sense if for every compact set \(K \subset \mathbb C\) with \(K \cap \overline{\Omega} = \varnothing\), \(K^c\) connected and for every function \(h\) continuous on \(K\) and holomorphic in \(K^\circ\), there exists a sequence \((\lambda_n)\) of nonnegative integers such that for every compact set \(L \subset \Omega\) there holds \[ \lim_{n\to\infty}{\sup_{\zeta \in L}{\sup_{z \in K}{| S_{\lambda_n}(f,\zeta)(z)-h(z)| }}} = 0\,. \] The author proves that the set \(U_1\) of all these universal functions is a dense \(G_\delta\) set in \(X\). In particular \(U_1 \neq \varnothing\). Also the author proves similar results on universal functions vanishing at \(\infty\) and which are smooth on the boundary of \(\Omega\) as well as results on universal functions in the strong sense (which means that \(K \cap \Omega = \varnothing\)). The main tools of the pro…
Mathematics Subject Classification ID: 30B30
Mathematics Subject Classification ID: 30B10
zbMATH DE Number: 5077606
zbMATH Keywords: universal functions
zbMATH Keywords: Taylor series
zbMATH Keywords: Baire's theorem
zbMATH Keywords: boundedness
zbMATH Keywords: smoothness
author: Nikolaos Tsirivas
MaRDI profile type: Publication
full work available at URL: s00013-006-1739-1
OpenAlex ID: W2005660576
DOI: 10.1007/S00013-006-1739-1
Recommended article: Boundary behaviour of functions which possess universal Taylor series
Recommended article: Boundary behavior of universal Taylor series and their derivatives
Recommended article: Boundary behaviour of universal Taylor series
Recommended article: Boundary behavior and Cesàro means of universal Taylor series
Recommended article: Universal Taylor series and summability
Recommended article: Smooth universal Taylor series on doubly connected domains
Recommended article: Universality properties of Taylor series inside the domain of holomorphy
Recommended article: Boundary behaviour of universal Taylor series on~multiply connected domains
Recommended article: A generalization of universal Taylor series in simply connected domains
Recommended article: Universal Taylor series in simply connected domains
description: scientific article; zbMATH DE number 5077606