Potential Theory/Complex Analysis
On the behaviour of power series in the absence of Hadamard–Ostrowski gaps
Comptes Rendus. Mathématique, Volume 351 (2013) no. 7-8, pp. 255-259.

We show that the partial sums (Snf)nN of a power series f with radius of convergence one tend to ∞ in capacity on (arbitrarily large) compact subsets of the complement of the closed unit disk, if f does not have so-called Hadamard–Ostrowski gaps. Regarding a recent result of Gardiner, this covers a large class of functions f holomorphic in the unit disk.

Nous montrons que les sommes partielles (Snf)nN dʼune série entière f de rayon de convergence 1 tendent vers ∞ en capacité sur les ensembles compacts (arbitrairement grands) du complémentaire du disque unité fermé si f ne contient pas de lacunes de Hadamard–Ostrowski. Tenant compte dʼun résultat récent de Gardiner, ceci couvre une grande classe de fonctions f holomorphes sur le disque unité.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.04.012
Kalmes, Thomas 1; Müller, Jürgen 1; Nieß, Markus 2

1 University of Trier, FB IV, Mathematics, Universitätsring, Trier, Germany
2 TU Clausthal, Institute of Mathematics, Erzstr. 1, Clausthal-Zellerfeld, Germany
@article{CRMATH_2013__351_7-8_255_0,
     author = {Kalmes, Thomas and M\"uller, J\"urgen and Nie{\ss}, Markus},
     title = {On the behaviour of power series in the absence of {Hadamard{\textendash}Ostrowski} gaps},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {255--259},
     publisher = {Elsevier},
     volume = {351},
     number = {7-8},
     year = {2013},
     doi = {10.1016/j.crma.2013.04.012},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.04.012/}
}
TY  - JOUR
AU  - Kalmes, Thomas
AU  - Müller, Jürgen
AU  - Nieß, Markus
TI  - On the behaviour of power series in the absence of Hadamard–Ostrowski gaps
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 255
EP  - 259
VL  - 351
IS  - 7-8
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.04.012/
DO  - 10.1016/j.crma.2013.04.012
LA  - en
ID  - CRMATH_2013__351_7-8_255_0
ER  - 
%0 Journal Article
%A Kalmes, Thomas
%A Müller, Jürgen
%A Nieß, Markus
%T On the behaviour of power series in the absence of Hadamard–Ostrowski gaps
%J Comptes Rendus. Mathématique
%D 2013
%P 255-259
%V 351
%N 7-8
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.04.012/
%R 10.1016/j.crma.2013.04.012
%G en
%F CRMATH_2013__351_7-8_255_0
Kalmes, Thomas; Müller, Jürgen; Nieß, Markus. On the behaviour of power series in the absence of Hadamard–Ostrowski gaps. Comptes Rendus. Mathématique, Volume 351 (2013) no. 7-8, pp. 255-259. doi : 10.1016/j.crma.2013.04.012. http://www.numdam.org/articles/10.1016/j.crma.2013.04.012/

[1] Baker, G.A.; Graves-Morris, P. Padé Approximants, Cambridge University Press, Cambridge, 1996

[2] Beise, P.; Meyrath, T.; Müller, J. Universality properties of Taylor series inside the domain of holomorphy, J. Math. Anal. Appl., Volume 383 (2011), pp. 234-238

[3] Gardiner, S.J. Existence of universal Taylor series for nonsimply connected domains, Constr. Approx., Volume 35 (2012), pp. 245-257

[4] Garnett, J.B.; Marshall, D.E. Harmonic Measure, Cambridge University Press, Cambridge, 2005

[5] Gehlen, W.; Luh, W.; Müller, J. On the existence of O-universal functions, Complex Var. Theory Appl., Volume 41 (2000), pp. 81-90

[6] Grosse-Erdmann, K.G. Universal families and hypercyclic operators, Bull. Am. Math. Soc. (N.S.), Volume 36 (1999), pp. 345-381

[7] Hille, E. Analytic Function Theory, Vol. II, Chelsea, New York, 1987

[8] Kahane, J.P. Baireʼs category theorem and trigonometric series, J. Anal. Math., Volume 80 (2000), pp. 143-182

[9] Melas, A. Universal functions on nonsimply connected domains, Ann. Inst. Fourier (Grenoble), Volume 51 (2001), pp. 1539-1551

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

[11] Meyer, B. On convergence in capacity, Bull. Aust. Math. Soc., Volume 14 (1976), pp. 1-5

[12] Müller, J.; Vlachou, V.; Yavrian, A. Universal overconvergence and Ostrowski-gaps, Bull. Lond. Math. Soc., Volume 38 (2006), pp. 597-606

[13] Ransford, T. Potential Theory in the Complex Plane, Cambridge University Press, 1995

Cited by Sources: