A holomorphic function on a simply connected domain is said to possess a universal Taylor series about a point in if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta outside (provided only that has connected complement). This paper shows that this property is not conformally invariant, and, in the case where is the unit disc, that such functions have extreme angular boundary behaviour.
On dit qu’une fonction , qui est holomorphe sur un domaine simplement connexe , possède une série universelle de Taylor autour d’un point de si tout polynôme sur tout compact en-dehors de peut être approximé par des sommes partielles de cette série (pourvu que le complémentaire de soit connexe). Cet article montre que cette propriété n’est pas invariante par transformation conforme et, dans le cas où est le disque unité, que ces fonctions ont un comportement extrême dans le sens des limites angulaires.
Accepted:
DOI: 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}, pages = {327--339}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {64}, number = {1}, year = {2014}, doi = {10.5802/aif.2849}, zbl = {06387276}, mrnumber = {3330551}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2849/} }
TY - JOUR TI - Universal Taylor series, conformal mappings and boundary behaviour JO - Annales de l'Institut Fourier PY - 2014 DA - 2014/// SP - 327 EP - 339 VL - 64 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2849/ UR - https://zbmath.org/?q=an%3A06387276 UR - https://www.ams.org/mathscinet-getitem?mr=3330551 UR - https://doi.org/10.5802/aif.2849 DO - 10.5802/aif.2849 LA - en ID - AIF_2014__64_1_327_0 ER -
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/articles/10.5802/aif.2849/
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