On the radial behavior of universal Taylor series
type: Item
instance of: scholarly article
title: On the radial behavior of universal Taylor series
zbMATH Open document ID: 1079.30002
author: G. Costakis
published in: Monatshefte für Mathematik
publication date: 2005-06-14T00:00:00Z
review text: In this paper, the author deals with universal Taylor series in the open unit disk \(D\). Here the universality is related to the overconvergence phenomenon. Let \(H(D)\) denote the Fréchet space of all holomorphic functions in \(D\). By definition, a function \(f \in H(D)\) is said to be a universal Taylor series in the sense of Nestoridis -- its set is denoted by \(U(D,0)\) -- if for every compact subset \(K \subset D^c\) with \(K^c\) connected and for every function \(h:K \to C\), continuous on \(K\) and holomorphic in \(K^0\), there exists a sequence \((n_j)\) of natural numbers such that \(\lim_{n \to \infty} \sup_{z \in K} | S_{n_j}(f,0)(z) - h(z)| = 0\), where \(S_n(f,0)\) denotes the \(n\)th partial sum of the Taylor development of \(f\) with center \(0\). Nestoridis proved in 1996 that \(U(D,0)\) is in fact a residual set in \(H(D)\), so improving earlier results by Luh, Chui and Parnes. Katsoprinakis, Papadimitrakis, Melas, Nestoridis and Papadoperakis have shown that universal series are not \((C,k)\) summable at every boundary point for every positive integer \(k\). Nevertheless, using an approximation theorem of Nersesjan together with a result due to Herzog about induced universality in \(G_\delta\) sets, the author proves that universal Taylor series can have a good radial behavior --namely, they can be Abel summable-- at some points of the unit circle. In fact, he is able to demonstrate the following stronger result: Let \(E\) be a closed and nowhere dense subset of the unit circle and \(S := \{rt: r \in [0,1)\), \(t \in E\}\), \(\varepsilon :[0,1) \to (0,+\infty )\) be a decreasing continuous function, and let \(g \in H(D)\). Consider the set \(A := \{f \in H(D): | f(z) - g(z)| < \varepsilon (| z| )\) \(\forall z \in S\}\). Then \(U(D,0) \cap A\) is residual in \(A\). An interesting question about the existence of universal Taylor series bounded on a sector is posed.
reviewed by: L. Bernal-González
Mathematics Subject Classification ID: 30B30
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 2175896
zbMATH Keywords: universal Taylor series
zbMATH Keywords: Baire's theorem
zbMATH Keywords: Mergelyan's approximation theorem
zbMATH Keywords: Nersesjan's approximation theorem
MaRDI profile type: Publication
full work available at URL: s00605-004-0276-x
OpenAlex ID: W1980155213
DOI: 10.1007/S00605-004-0276-X
description: scientific article; zbMATH DE number 2175896