Universal Taylor series, conformal mappings and boundary behaviour
Annales de l'Institut Fourier, Volume 64 (2014) no. 1, p. 327-339
A holomorphic function f on a simply connected domain Ω is said to possess a universal Taylor series about a point in Ω if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta K outside Ω (provided only that K has connected complement). This paper shows that this property is not conformally invariant, and, in the case where Ω is the unit disc, that such functions have extreme angular boundary behaviour.
On dit qu’une fonction f, qui est holomorphe sur un domaine simplement connexe Ω, possède une série universelle de Taylor autour d’un point de Ω si tout polynôme sur tout compact K en-dehors de Ω peut être approximé par des sommes partielles de cette série (pourvu que le complémentaire de K soit connexe). Cet article montre que cette propriété n’est pas invariante par transformation conforme et, dans le cas où Ω est le disque unité, que ces fonctions ont un comportement extrême dans le sens des limites angulaires.
Received : 2012-03-02
Accepted : 2012-12-14
DOI : https://doi.org/10.5802/aif.2849
Classification:  30K05,  30B30,  30E10,  31A05
Keywords: Universal Taylor series, conformal mappings, angular boundary behaviour.
@article{AIF_2014__64_1_327_0,
     author = {Gardiner, Stephen J.},
     title = {Universal Taylor series, conformal mappings and boundary behaviour},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {1},
     year = {2014},
     pages = {327-339},
     doi = {10.5802/aif.2849},
     zbl = {06387276},
     mrnumber = {3330551},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2014__64_1_327_0}
}
Gardiner, Stephen J. Universal Taylor series, conformal mappings and boundary behaviour. Annales de l'Institut Fourier, Volume 64 (2014) no. 1, pp. 327-339. doi : 10.5802/aif.2849. https://aif.centre-mersenne.org/item/AIF_2014__64_1_327_0/

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