Boundary behaviour of universal Taylor series

Gardiner, Stephen J.; Khavinson, Dmitry

Comptes Rendus. Mathématique, Tome 352 (2014) no. 2, pp. 99-103.

Une série entière qui converge sur le disque unité $\mathbb{D}$ est appelée universelle si tout polynôme peut être approximé, sur tout compact de $\mathbb{C}\backslash \mathbb{D}$ ayant un complémentaire connexe, par ses sommes partielles. Cet article montre que ces séries croissent fortement et possèdent une propriété du type Picard près de chaque point de la frontière.

A power series that converges on the unit disc $\mathbb{D}$ is called universal if its partial sums approximate arbitrary polynomials on arbitrary compacta in $\mathbb{C}\backslash \mathbb{D}$ that have connected complement. This paper shows that such series grow strongly and possess a Picard-type property near each boundary point.

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