Universal Taylor series, conformal mappings and boundary behaviour  [ Séries de Taylor universelles, transformations conformes et comportement à la frontière ]
Annales de l'Institut Fourier, Tome 64 (2014) no. 1, p. 327-339
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.
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.
DOI : https://doi.org/10.5802/aif.2849
Classification:  30K05,  30B30,  30E10,  31A05
Mots clés: Séries de Taylor universelles, transformations conformes, comportement angulaire à la frontière.
@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},
     mrnumber = {3330551},
     zbl = {06387276},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2014__64_1_327_0}
}
Gardiner, Stephen J. Universal Taylor series, conformal mappings and boundary behaviour. Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 327-339. doi : 10.5802/aif.2849. http://www.numdam.org/item/AIF_2014__64_1_327_0/

[1] Armitage, D. H.; Costakis, G. Boundary behavior of universal Taylor series and their derivatives, Constr. Approx., Tome 24 (2006), pp. 1-15 | Article | MR 2217523 | Zbl 1098.30003

[2] Armitage, D. H.; Gardiner, S. J. Classical Potential Theory, Springer, London (2001) | MR 1801253 | Zbl 0972.31001

[3] Barth, K. F.; Rippon, P. J. Extensions of a theorem of Valiron, Bull. London Math. Soc., Tome 38 (2006), pp. 815-824 | Article | MR 2268366 | Zbl 1115.30037

[4] Bayart, F. Boundary behavior and Cesàro means of universal Taylor series, Rev. Mat. Complut., Tome 19 (2006), pp. 235-247 | Article | MR 2219831 | Zbl 1103.30003

[5] Bernal-González, L.; Bonilla, A.; Calderón-Moreno, M. C.And; Prado-Bassas, J. A. Universal Taylor series with maximal cluster sets, Rev. Mat. Iberoam., Tome 25 (2009), pp. 757-780 | Article | MR 2569553 | Zbl 1186.30003

[6] Brelot, M.; Doob, J. L. Limites angulaires et limites fines, Ann. Inst. Fourier (Grenoble), Tome 13 (1963) no. 2, pp. 395-415 | Article | Numdam | MR 196107 | Zbl 0132.33902

[7] Costakis, G. On the radial behavior of universal Taylor series, Monatsh. Math., Tome 145 (2005), pp. 11-17 | Article | MR 2134476 | Zbl 1079.30002

[8] Costakis, G. Which maps preserve universal functions?, Oberwolfach Rep., Tome 6 (2008), pp. 328-331

[9] Costakis, G.; Melas, A. On the range of universal functions, Bull. London Math. Soc., Tome 32 (2000), pp. 458-464 | Article | MR 1760810 | Zbl 1023.30003

[10] Doob, J. L. Classical Potential Theory and its Probabilistic Counterpart, Springer, New York (1984) | MR 731258 | Zbl 0990.31001

[11] Gardiner, S. J. Boundary behaviour of functions which possess universal Taylor series (Bull. London Math. Soc., to appear) | MR 3033966 | Zbl 1272.30081

[12] Lelong-Ferrand, J. Étude au voisinage de la frontière des fonctions subharmoniques positives dans un demi-espace, Ann. Sci. École Norm. Sup. (3), Tome 66 (1949), pp. 125-159 | Numdam | MR 31603 | Zbl 0033.37301

[13] Melas, A. On the growth of universal functions, J. Anal. Math., Tome 82 (2000), pp. 1-20 | Article | MR 1799655 | Zbl 0973.30002

[14] Melas, A.; Nestoridis, V. Universality of Taylor series as a generic property of holomorphic functions, Adv. Math., Tome 157 (2001), pp. 138-176 | Article | MR 1813429 | Zbl 0985.30023

[15] Melas, A.; Nestoridis, V.; Papadoperakis, I. Growth of coefficients of universal Taylor series and comparison of two classes of functions, J. Anal. Math., Tome 73 (1997), pp. 187-202 | Article | MR 1616485 | Zbl 0894.30003

[16] Müller, J.; Vlachou, V.; Yavrian, A. Universal overconvergence and Ostrowski-gaps, Bull. London Math. Soc., Tome 38 (2006), pp. 597-606 | Article | MR 2250752 | Zbl 1099.30001

[17] Nestoridis, V. Universal Taylor series, Ann. Inst. Fourier (Grenoble), Tome 46 (1996), pp. 1293-1306 | Article | Numdam | MR 1427126 | Zbl 0865.30001

[18] Nestoridis, V. An extension of the notion of universal Taylor series, Computational methods and function theory 1997 (Nicosia), World Sci. Publ., River Edge, NJ (Ser. Approx. Decompos.) Tome 11 (1999), pp. 421-430 | MR 1700365 | Zbl 0942.30003

[19] Pommerenke, C. Boundary Behaviour of Conformal Maps, Springer, Berlin (1992) | MR 1217706 | Zbl 0762.30001

[20] Ransford, T. Potential Theory in the Complex Plane, Cambridge Univ. Press (1995) | MR 1334766 | Zbl 0828.31001