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Comptes Rendus Mathématique
Volume 348, n° 9-10
pages 521-524 (mai 2010)
Doi : 10.1016/j.crma.2010.03.003
Received : 28 January 2010 ;  accepted : 3 Mars 2010
Universal Taylor series for non-simply connected domains
Séries universelles de Taylor pour les domaines non-simplement connexes
 

Stephen J. Gardiner , Nikolaos Tsirivas
School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland 

Abstract

It is known that, for any simply connected proper subdomain Ω of the complex plane and any point ζ in Ω , there are holomorphic functions on Ω that have “universal” Taylor series expansions about ζ ; that is, partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compacta in   that have connected complement. This note shows that this phenomenon can break down for non-simply connected domains Ω , even when   is compact. This answers a question of Melas and disproves a conjecture of Müller, Vlachou and Yavrian.

The full text of this article is available in PDF format.
Résumé

Il est connu que, pour un sous-domaine propre simplement connexe Ω du plan complexe et un point quelconque ζ de Ω , il y a des fonctions holomorphes sur Ω qui possèdent des séries de Taylor « universelles » autour de ζ ; c’est-à-dire tout polynôme peut être approximé, sur tout compact de   ayant un complémentaire connexe, par les sommes partielles de la série de Taylor. Cette note montre que ce résultat n’est plus vrai en général pour les domaines non-simplement connexes Ω , même lorsque   est compact. Cela répond à une question de Melas et réfute une conjecture de Müller, Vlachou et Yavrian.

The full text of this article is available in PDF format.

 This research was supported by Science Foundation Ireland under Grant 09/RFP/MTH2149, and is also part of the programme of the ESF Network “Harmonic and Complex Analysis and Applications” (HCAA).



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