# Universal Taylor series, conformal mappings and boundary behaviour

Stephen J. Gardiner^{[1]}

- [1] School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland.

Annales de l’institut Fourier (2014)

- Volume: 64, Issue: 1, page 327-339
- ISSN: 0373-0956

## Access Full Article

top## Abstract

top## How to cite

topGardiner, Stephen J.. "Universal Taylor series, conformal mappings and boundary behaviour." Annales de l’institut Fourier 64.1 (2014): 327-339. <http://eudml.org/doc/275643>.

@article{Gardiner2014,

abstract = {A holomorphic function $f$ on a simply connected domain $\Omega $ is said to possess a universal Taylor series about a point in $\Omega $ if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta $K$ outside $\Omega $ (provided only that $K$ has connected complement). This paper shows that this property is not conformally invariant, and, in the case where $\Omega $ is the unit disc, that such functions have extreme angular boundary behaviour.},

affiliation = {School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland.},

author = {Gardiner, Stephen J.},

journal = {Annales de l’institut Fourier},

keywords = {Universal Taylor series; conformal mappings; angular boundary behaviour; universal Taylor series; angular boundary behavior},

language = {eng},

number = {1},

pages = {327-339},

publisher = {Association des Annales de l’institut Fourier},

title = {Universal Taylor series, conformal mappings and boundary behaviour},

url = {http://eudml.org/doc/275643},

volume = {64},

year = {2014},

}

TY - JOUR

AU - Gardiner, Stephen J.

TI - Universal Taylor series, conformal mappings and boundary behaviour

JO - Annales de l’institut Fourier

PY - 2014

PB - Association des Annales de l’institut Fourier

VL - 64

IS - 1

SP - 327

EP - 339

AB - A holomorphic function $f$ on a simply connected domain $\Omega $ is said to possess a universal Taylor series about a point in $\Omega $ if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta $K$ outside $\Omega $ (provided only that $K$ has connected complement). This paper shows that this property is not conformally invariant, and, in the case where $\Omega $ is the unit disc, that such functions have extreme angular boundary behaviour.

LA - eng

KW - Universal Taylor series; conformal mappings; angular boundary behaviour; universal Taylor series; angular boundary behavior

UR - http://eudml.org/doc/275643

ER -

## References

top- D. H. Armitage, G. Costakis, Boundary behavior of universal Taylor series and their derivatives, Constr. Approx. 24 (2006), 1-15 Zbl1098.30003MR2217523
- D. H. Armitage, S. J. Gardiner, Classical Potential Theory, (2001), Springer, London Zbl0972.31001MR1801253
- K. F. Barth, P. J. Rippon, Extensions of a theorem of Valiron, Bull. London Math. Soc. 38 (2006), 815-824 Zbl1115.30037MR2268366
- F. Bayart, Boundary behavior and Cesàro means of universal Taylor series, Rev. Mat. Complut. 19 (2006), 235-247 Zbl1103.30003MR2219831
- L. Bernal-González, A. Bonilla, M. C.and Calderón-Moreno, J. A. Prado-Bassas, Universal Taylor series with maximal cluster sets, Rev. Mat. Iberoam. 25 (2009), 757-780 Zbl1186.30003MR2569553
- M. Brelot, J. L. Doob, Limites angulaires et limites fines, Ann. Inst. Fourier (Grenoble) 13 (1963), 395-415 Zbl0132.33902MR196107
- G. Costakis, On the radial behavior of universal Taylor series, Monatsh. Math. 145 (2005), 11-17 Zbl1079.30002MR2134476
- G. Costakis, Which maps preserve universal functions?, Oberwolfach Rep. 6 (2008), 328-331
- G. Costakis, A. Melas, On the range of universal functions, Bull. London Math. Soc. 32 (2000), 458-464 Zbl1023.30003MR1760810
- J. L. Doob, Classical Potential Theory and its Probabilistic Counterpart, (1984), Springer, New York Zbl0990.31001MR731258
- S. J. Gardiner, Boundary behaviour of functions which possess universal Taylor series Zbl1272.30081MR3033966
- J. Lelong-Ferrand, Étude au voisinage de la frontière des fonctions subharmoniques positives dans un demi-espace, Ann. Sci. École Norm. Sup. (3) 66 (1949), 125-159 Zbl0033.37301MR31603
- A. Melas, On the growth of universal functions, J. Anal. Math. 82 (2000), 1-20 Zbl0973.30002MR1799655
- A. Melas, V. Nestoridis, Universality of Taylor series as a generic property of holomorphic functions, Adv. Math. 157 (2001), 138-176 Zbl0985.30023MR1813429
- A. Melas, V. Nestoridis, I. Papadoperakis, Growth of coefficients of universal Taylor series and comparison of two classes of functions, J. Anal. Math. 73 (1997), 187-202 Zbl0894.30003MR1616485
- J. Müller, V. Vlachou, A. Yavrian, Universal overconvergence and Ostrowski-gaps, Bull. London Math. Soc. 38 (2006), 597-606 Zbl1099.30001MR2250752
- V. Nestoridis, Universal Taylor series, Ann. Inst. Fourier (Grenoble) 46 (1996), 1293-1306 Zbl0865.30001MR1427126
- V. Nestoridis, An extension of the notion of universal Taylor series, Computational methods and function theory 1997 (Nicosia) 11 (1999), 421-430, World Sci. Publ., River Edge, NJ Zbl0942.30003MR1700365
- C. Pommerenke, Boundary Behaviour of Conformal Maps, (1992), Springer, Berlin Zbl0762.30001MR1217706
- T. Ransford, Potential Theory in the Complex Plane, (1995), Cambridge Univ. Press Zbl0828.31001MR1334766

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.