Universal Taylor series for non-simply connected domains

Stephen J. Gardiner; Nikolaos Tsirivas

doi:10.1016/j.crma.2010.03.003

Complex Analysis

Universal Taylor series for non-simply connected domains
[Séries universelles de Taylor pour les domaines non-simplement connexes]

Présenté par : Jean-Pierre Kahane

Stephen J. Gardiner ¹ ; Nikolaos Tsirivas ¹

¹ School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 521-524.

Résumé
Abstract

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 $C \ Ω$ 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 $C \ Ω$ est compact. Cela répond à une question de Melas et réfute une conjecture de Müller, Vlachou et Yavrian.

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 $C \ Ω$ that have connected complement. This note shows that this phenomenon can break down for non-simply connected domains Ω, even when $C \ Ω$ is compact. This answers a question of Melas and disproves a conjecture of Müller, Vlachou and Yavrian.

Reçu le : 2010-01-28
Accepté le : 2010-03-03
Publié le : 2010-03-29

DOI : 10.1016/j.crma.2010.03.003

Affiliations des auteurs :

Stephen J. Gardiner ¹ ; Nikolaos Tsirivas ¹

¹ School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

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Stephen J. Gardiner; Nikolaos Tsirivas. Universal Taylor series for non-simply connected domains. Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 521-524. doi : 10.1016/j.crma.2010.03.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.03.003/

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Cité par Sources :

^☆ 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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