Universal Taylor series for non-simply connected domains

Article history: Received 28 January 2010 Accepted 3 March 2010 Available online 29 March 2010 Presented by Jean-Pierre Kahane 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 Muller, Vlachou and Yavrian. © 2010 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved. r e s u m e 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 possedent des series de Taylor «universelles » autour de ζ ; c’est-a-dire tout polynome peut etre approxime, sur tout compact de C\Ω ayant un complementaire connexe, par les sommes partielles de la serie de Taylor. Cette note montre que ce resultat n’est plus vrai en general pour les domaines non-simplement connexes Ω , meme lorsque C\Ω est compact. Cela repond a une question de Melas et refute une conjecture de Muller, Vlachou et Yavrian. © 2010 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

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