Universal Taylor series

Nestoridis, Vassili

Annales de l'Institut Fourier, Tome 46 (1996) no. 5, pp. 1293-1306.

Nous améliorons un résultat de Chui et Parnes et nous démontrons que les séries de Taylor universelles forment un sous-espace $G_{\delta }$-dense de l’espace de fonctions holomorphes définies sur le disque unité ouvert. Nous utilisons ce résultat pour répondre à une question de S.K. Pichorides sur l’ensemble limite des séries de Taylor. Nous étudions aussi quelques propriétés des séries de Taylor universelles; en particulier, ce sont des séries trigonométriques universelles au sens de Menchoff.

We strengthen a result of Chui and Parnes and we prove that the set of universal Taylor series is a $G_{\delta }$-dense subset of the space of holomorphic functions defined in the open unit disc. Our result provides the answer to a question stated by S.K. Pichorides concerning the limit set of Taylor series. Moreover, we study some properties of universal Taylor series and show, in particular, that they are trigonometric series in the sense of D. Menchoff.

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