Universal Taylor series, conformal mappings and boundary behaviour
[Séries de Taylor universelles, transformations conformes et comportement à la frontière]
Gardiner, Stephen J.1
1 School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland.
Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 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 : 10.5802/aif.2849
Classification : 30K05, 30B30, 30E10, 31A05
Keywords: Universal Taylor series, conformal mappings, angular boundary behaviour.
Mot clés : Séries de Taylor universelles, transformations conformes, comportement angulaire à la frontière.
Gardiner, Stephen J. 1

1 School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland. 
@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 = {http://www.numdam.org/articles/10.5802/aif.2849/}
}
TY  - JOUR
AU  - Gardiner, Stephen J.
TI  - Universal Taylor series, conformal mappings and boundary behaviour
JO  - Annales de l'Institut Fourier
PY  - 2014
SP  - 327
EP  - 339
VL  - 64
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2849/
DO  - 10.5802/aif.2849
LA  - en
ID  - AIF_2014__64_1_327_0
ER  - 
%0 Journal Article
%A Gardiner, Stephen J.
%T Universal Taylor series, conformal mappings and boundary behaviour
%J Annales de l'Institut Fourier
%D 2014
%P 327-339
%V 64
%N 1
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2849/
%R 10.5802/aif.2849
%G en
%F 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/articles/10.5802/aif.2849/

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

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

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

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

[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., Volume 25 (2009), pp. 757-780 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :