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

Abstract

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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.

How to cite

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Gardiner, 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

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  1. D. H. Armitage, G. Costakis, Boundary behavior of universal Taylor series and their derivatives, Constr. Approx. 24 (2006), 1-15 Zbl1098.30003MR2217523
  2. D. H. Armitage, S. J. Gardiner, Classical Potential Theory, (2001), Springer, London Zbl0972.31001MR1801253
  3. K. F. Barth, P. J. Rippon, Extensions of a theorem of Valiron, Bull. London Math. Soc. 38 (2006), 815-824 Zbl1115.30037MR2268366
  4. F. Bayart, Boundary behavior and Cesàro means of universal Taylor series, Rev. Mat. Complut. 19 (2006), 235-247 Zbl1103.30003MR2219831
  5. 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
  6. M. Brelot, J. L. Doob, Limites angulaires et limites fines, Ann. Inst. Fourier (Grenoble) 13 (1963), 395-415 Zbl0132.33902MR196107
  7. G. Costakis, On the radial behavior of universal Taylor series, Monatsh. Math. 145 (2005), 11-17 Zbl1079.30002MR2134476
  8. G. Costakis, Which maps preserve universal functions?, Oberwolfach Rep. 6 (2008), 328-331 
  9. G. Costakis, A. Melas, On the range of universal functions, Bull. London Math. Soc. 32 (2000), 458-464 Zbl1023.30003MR1760810
  10. J. L. Doob, Classical Potential Theory and its Probabilistic Counterpart, (1984), Springer, New York Zbl0990.31001MR731258
  11. S. J. Gardiner, Boundary behaviour of functions which possess universal Taylor series Zbl1272.30081MR3033966
  12. 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
  13. A. Melas, On the growth of universal functions, J. Anal. Math. 82 (2000), 1-20 Zbl0973.30002MR1799655
  14. A. Melas, V. Nestoridis, Universality of Taylor series as a generic property of holomorphic functions, Adv. Math. 157 (2001), 138-176 Zbl0985.30023MR1813429
  15. 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
  16. J. Müller, V. Vlachou, A. Yavrian, Universal overconvergence and Ostrowski-gaps, Bull. London Math. Soc. 38 (2006), 597-606 Zbl1099.30001MR2250752
  17. V. Nestoridis, Universal Taylor series, Ann. Inst. Fourier (Grenoble) 46 (1996), 1293-1306 Zbl0865.30001MR1427126
  18. 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
  19. C. Pommerenke, Boundary Behaviour of Conformal Maps, (1992), Springer, Berlin Zbl0762.30001MR1217706
  20. T. Ransford, Potential Theory in the Complex Plane, (1995), Cambridge Univ. Press Zbl0828.31001MR1334766

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