Nous montrons que les sommes partielles 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.
We show that the partial sums 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.
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Publié le :
Thomas Kalmes 1 ; Jürgen Müller 1 ; Markus Nieß 2
@article{CRMATH_2013__351_7-8_255_0, author = {Thomas Kalmes and J\"urgen M\"uller and Markus Nie{\ss}}, 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}, }
TY - JOUR AU - Thomas Kalmes AU - Jürgen Müller AU - Markus Nieß 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 DO - 10.1016/j.crma.2013.04.012 LA - en ID - CRMATH_2013__351_7-8_255_0 ER -
Thomas Kalmes; Jürgen Müller; Markus Nieß. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.04.012/
[1] Padé Approximants, Cambridge University Press, Cambridge, 1996
[2] Universality properties of Taylor series inside the domain of holomorphy, J. Math. Anal. Appl., Volume 383 (2011), pp. 234-238
[3] Existence of universal Taylor series for nonsimply connected domains, Constr. Approx., Volume 35 (2012), pp. 245-257
[4] Harmonic Measure, Cambridge University Press, Cambridge, 2005
[5] On the existence of O-universal functions, Complex Var. Theory Appl., Volume 41 (2000), pp. 81-90
[6] Universal families and hypercyclic operators, Bull. Am. Math. Soc. (N.S.), Volume 36 (1999), pp. 345-381
[7] Analytic Function Theory, Vol. II, Chelsea, New York, 1987
[8] Baireʼs category theorem and trigonometric series, J. Anal. Math., Volume 80 (2000), pp. 143-182
[9] Universal functions on nonsimply connected domains, Ann. Inst. Fourier (Grenoble), Volume 51 (2001), pp. 1539-1551
[10] Universality of Taylor series as a generic property of holomorphic functions, Adv. Math., Volume 157 (2001), pp. 138-176
[11] On convergence in capacity, Bull. Aust. Math. Soc., Volume 14 (1976), pp. 1-5
[12] Universal overconvergence and Ostrowski-gaps, Bull. Lond. Math. Soc., Volume 38 (2006), pp. 597-606
[13] Potential Theory in the Complex Plane, Cambridge University Press, 1995
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