Complex analysis/Functional analysis
On the boundary behaviour of derivatives of functions in the disc algebra
Comptes Rendus. Mathématique, Volume 356 (2018) no. 7, pp. 732-736.

We provide with a simple and explicit construction of a function in the disc algebra, whose derivatives enjoy a disjoint universal property near the boundary. The set of functions with such property is topologically generic, densely lineable, and spaceable.

On présente une construction simple et explicite d'une fonction de l'algèbre du disque dont les dérivés possèdent des propriétés d'universalité disjointe au bord. L'ensemble des fonctions ayant une telle propriété est topologiquement générique et contient un sous-espace dense et un sous-espace fermé de dimension infinie.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.05.015
Charpentier, Stéphane 1; Nestoridis, Vassili 2

1 Institut de mathématiques de Marseille, UMR 7353, Aix-Marseille Université, Technopôle Château-Gombert, 39, rue Frédéric-Joliot-Curie, 13453 Marseille cedex 13, France
2 Department of Mathematics, National and Kapodistrian University of Athens, 15784 Athens, Greece
@article{CRMATH_2018__356_7_732_0,
     author = {Charpentier, St\'ephane and Nestoridis, Vassili},
     title = {On the boundary behaviour of derivatives of functions in the disc algebra},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {732--736},
     publisher = {Elsevier},
     volume = {356},
     number = {7},
     year = {2018},
     doi = {10.1016/j.crma.2018.05.015},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2018.05.015/}
}
TY  - JOUR
AU  - Charpentier, Stéphane
AU  - Nestoridis, Vassili
TI  - On the boundary behaviour of derivatives of functions in the disc algebra
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 732
EP  - 736
VL  - 356
IS  - 7
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2018.05.015/
DO  - 10.1016/j.crma.2018.05.015
LA  - en
ID  - CRMATH_2018__356_7_732_0
ER  - 
%0 Journal Article
%A Charpentier, Stéphane
%A Nestoridis, Vassili
%T On the boundary behaviour of derivatives of functions in the disc algebra
%J Comptes Rendus. Mathématique
%D 2018
%P 732-736
%V 356
%N 7
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2018.05.015/
%R 10.1016/j.crma.2018.05.015
%G en
%F CRMATH_2018__356_7_732_0
Charpentier, Stéphane; Nestoridis, Vassili. On the boundary behaviour of derivatives of functions in the disc algebra. Comptes Rendus. Mathématique, Volume 356 (2018) no. 7, pp. 732-736. doi : 10.1016/j.crma.2018.05.015. http://www.numdam.org/articles/10.1016/j.crma.2018.05.015/

[1] Aron, R.M.; Bernal-González, L.; Pellegrino, D.M.; Seoane Sepúlveda, J.B. Lineability: The Search for Linearity in Mathematics, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, USA, 2016

[2] Bagemihl, F.; Seidel, W. Some boundary properties of analytic functions, Math. Z., Volume 61 (1954) no. 1, pp. 186-199

[3] Bayart, F. Universal radial limits of holomorphic functions, Glasg. Math. J., Volume 47 (2005) no. 2, pp. 261-267

[4] Bayart, F.; Grosse-Erdmann, K.-G.; Nestoridis, V.; Papadimitropoulos, C. Abstract theory of universal series and applications, Proc. Lond. Math. Soc., Volume 96 (2008), pp. 417-463

[5] Bernal-González, L. Disjoint hypercyclic operators, Stud. Math., Volume 182 (2007) no. 2, pp. 113-131

[6] Bernal-González, L.; Pellegrino, D.; Seoane-Sepúlveda, J.B. Linear subsets of nonlinear sets in topological vector spaces, Bull. Amer. Math. Soc., Volume 51 (2014) no. 1, pp. 71-130

[7] Bès, J.; Peris, A. Disjointness in hypercyclicity, J. Math. Anal. Appl., Volume 336 (2007) no. 1, pp. 297-315

[8] Dupain, Y. Extension à la dimension n d'un théorème de Ortel et Schneider, Math. Z., Volume 206 (1991) no. 1, pp. 71-80

[9] Gardiner, S.J. Universal Taylor series, conformal mappings and boundary behaviour, Ann. Inst. Fourier (Grenoble), Volume 64 (2014) no. 1, pp. 327-339

[10] Hatziafratis, T.; Kioulafa, K.; Nestoridis, V. On Bergman type spaces of holomorphic functions and the density, in these spaces, of certain classes of singular functions, Complex Var. Elliptic Equ. (2017)

[11] Kahane, J.-P.; Katznelson, Y. Sur le comportement radial des fonctions analytiques, C. R. Acad. Sci. Paris, Ser. A, Volume 272 (1971), pp. 718-719

[12] Krantz, S. Smoothness to the boundary of biholomorphic mappings: an overview (Jarosz, K., ed.), Contemporary Mathematics, vol. 645, American Mathematical Society, Providence, Rhode Island, 2015, pp. 179-190 | DOI

[13] Menet, Q. Hypercyclic subspaces and weighted shifts, Adv. Math., Volume 255 (2014), pp. 305-337

[14] Nestoridis, V. Universal Taylor series, Ann. Inst. Fourier (Grenoble), Volume 46 (1996), pp. 1293-1306

[15] Nestoridis, V. Domains of holomorphy, Ann. Math. Qué., Volume 42 (2018) no. 1, pp. 101-105

[16] Painlevé, P. Sur les lignes singulières des fonctions analytiques, 1887 (PhD thesis)

[17] Siskaki, M. Boundedness of derivatives and anti-derivatives of holomorphic functions as a rare phenomenon, J. Math. Anal. Appl., Volume 462 (2018) no. 2, pp. 1073-1086

Cited by Sources: