[Séries universelles de Taylor pour les domaines non-simplement connexes]
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.
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.
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@article{CRMATH_2010__348_9-10_521_0,
author = {Stephen J. Gardiner and Nikolaos Tsirivas},
title = {Universal {Taylor} series for non-simply connected domains},
journal = {Comptes Rendus. Math\'ematique},
pages = {521--524},
publisher = {Elsevier},
volume = {348},
number = {9-10},
year = {2010},
doi = {10.1016/j.crma.2010.03.003},
language = {en},
}
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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☆ 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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