Comptes Rendus
no. 2
Potential theory/Complex analysis
Boundary behaviour of universal Taylor series

Presented by: Jean-Pierre Kahane

Stephen J. Gardiner1; Dmitry Khavinson2
1 School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
2 Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA
Comptes Rendus. Mathématique, Volume 352 (2014) no. 2, pp. 99-103.

A power series that converges on the unit disc D is called universal if its partial sums approximate arbitrary polynomials on arbitrary compacta in C\D that have connected complement. This paper shows that such series grow strongly and possess a Picard-type property near each boundary point.

Une série entière qui converge sur le disque unité D est appelée universelle si tout polynôme peut être approximé, sur tout compact de C\D ayant un complémentaire connexe, par ses sommes partielles. Cet article montre que ces séries croissent fortement et possèdent une propriété du type Picard près de chaque point de la frontière.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.12.008

Stephen J. Gardiner 1; Dmitry Khavinson 2

1 School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
2 Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA 
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Stephen J. Gardiner; Dmitry Khavinson. Boundary behaviour of universal Taylor series. Comptes Rendus. Mathématique, Volume 352 (2014) no. 2, pp. 99-103. doi : 10.1016/j.crma.2013.12.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.12.008/

[1] L.V. Ahlfors An extension of Schwarzʼs lemma, Trans. Amer. Math. Soc., Volume 43 (1938), pp. 359-364

[2] D.H. Armitage; G. Costakis Boundary behavior of universal Taylor series and their derivatives, Constr. Approx., Volume 24 (2006), pp. 1-15

[3] D.H. Armitage; S.J. Gardiner Classical Potential Theory, Springer, London, 2001

[4] F. Bayart Boundary behavior and Cesàro means of universal Taylor series, Rev. Mat. Complut., Volume 19 (2006), pp. 235-247

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

[6] L. Bernal-González; A. Bonilla; M.C. Calderón-Moreno; J.A. Prado-Bassas Universal Taylor series with maximal cluster sets, Rev. Mat. Iberoam., Volume 25 (2009), pp. 757-780

[7] G. Costakis On the radial behavior of universal Taylor series, Monatshefte Math., Volume 145 (2005), pp. 11-17

[8] G. Costakis; A. Melas On the range of universal functions, Bull. Lond. Math. Soc., Volume 32 (2000), pp. 458-464

[9] Y. Domar On the existence of a largest subharmonic minorant of a given function, Ark. Mat., Volume 3 (1957), pp. 429-440

[10] S.J. Gardiner Boundary behaviour of functions which possess universal Taylor series, Bull. Lond. Math. Soc., Volume 45 (2013), pp. 191-199

[11] S.J. Gardiner Universal Taylor series, conformal mappings and boundary behaviour, Ann. Inst. Fourier, Volume 63 (2013) (in press)

[12] P.M. Gauthier; I. Tamptse Universal overconvergence of homogeneous expansions of harmonic functions, Analysis, Volume 26 (2006), pp. 287-293

[13] M. Manolaki Universal polynomial expansions of harmonic functions, Potential Anal., Volume 38 (2013), pp. 985-1000

[14] A. Melas On the growth of universal functions, J. Anal. Math., Volume 82 (2000), pp. 1-20

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

[16] A. Melas; V. Nestoridis; I. Papadoperakis Growth of coefficients of universal Taylor series and comparison of two classes of functions, J. Anal. Math., Volume 73 (1997), pp. 187-202

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

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

[19] V. Nestoridis An extension of the notion of universal Taylor series, Nicosia, 1997 (Ser. Approx. Decompos.), Volume vol. 11, World Sci. Publ., River Edge, NJ (1999), pp. 421-430

[20] N. Sjöberg Sur les minorantes sousharmoniques dʼune fonction donnée, 9e$ {9}^{\mathrm{e}}$ Congr. des mathématiques scandinaves, 1939, pp. 309-319

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☆ The first author was supported by Science Foundation Ireland under Grant 09/RFP/MTH2149, and the second author by NSF grant DMS 0855597.

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