Universal Taylor series on arbitrary planar domains

Vassili Nestoridis; Christos Papachristodoulos

doi:10.1016/j.crma.2009.02.007

Complex Analysis/Harmonic Analysis

Universal Taylor series on arbitrary planar domains

Presented by: Jean-Pierre Kahane

Vassili Nestoridis ¹; Christos Papachristodoulos ¹

¹ Department of Mathematics, Panepistemiopolis, 157-84, Athenes, Greece

Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 363-367.

Abstract
Résumé

Let $Ω \subset C$ , $Ω \neq C$ be any domain and $ζ \in Ω$ . Let $R = dist (ζ, Ω^{c}) \in (0, + \infty)$ and $C (ζ, R) = {z \in C : | ζ - z | = R}$ . We set $J (Ω, ζ) = Ω^{c} \cap C (ζ, R)$ . Then there exists $f \in H (Ω)$ , such that the sequence $S_{N} (f, ζ) (z) = \sum_{n = 0}^{N} \frac{f^{(n)} (ζ)}{n!} {(z - ζ)}^{n}$ , $N = 0, 1, \dots$ , approximates any polynomial uniformly on each compact set $K \subset J (Ω, ζ)$ with $C ∖ K$ connected. This property of $f \in H (Ω)$ is topologically and algebraically generic.

Soit $Ω \subset C$ un domaine, avec $Ω \neq C$ . Soient aussi $ζ \in Ω$ , $R = dist (ζ, Ω^{c}) \in (0, + \infty)$ et $C (ζ, R) = {z \in C : | ζ - z | = R}$ . On pose $J (Ω, ζ) = Ω^{c} \cap C (ζ, R)$ . Alors il existe $f \in H (Ω)$ telle que la suite $S_{N} (f, ζ) (z) = \sum_{n = 0}^{N} \frac{f^{(n)} (ζ)}{n!} {(z - ζ)}^{n}$ , $N = 0, 1, \dots$ , approche tout polynôme uniformément sur tout compact $K \subset J (Ω, ζ)$ ne séparant pas le plan. Le phénomène est topologiquement et algébriquement générique.

Received: 2009-01-19
Accepted: 2009-01-27
Published online: 2009-03-06

DOI: 10.1016/j.crma.2009.02.007

Author's affiliations:

Vassili Nestoridis ¹; Christos Papachristodoulos ¹

¹ Department of Mathematics, Panepistemiopolis, 157-84, Athenes, Greece

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     title = {Universal {Taylor} series on arbitrary planar domains},
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Vassili Nestoridis; Christos Papachristodoulos. Universal Taylor series on arbitrary planar domains. Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 363-367. doi : 10.1016/j.crma.2009.02.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.02.007/

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