Universal Taylor series for non-simply connected domains

Stephen J. Gardiner; Nikolaos Tsirivas

doi:10.1016/j.crma.2010.03.003

Complex Analysis

Universal Taylor series for non-simply connected domains

Presented by: Jean-Pierre Kahane

Stephen J. Gardiner ¹; Nikolaos Tsirivas ¹

¹ School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 521-524.

Abstract (Original language)
Résumé

It is known that, for any simply connected proper subdomain Ω of the complex plane and any point ζ in Ω, there are holomorphic functions on Ω that have “universal” Taylor series expansions about ζ; that is, partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compacta in $C \ Ω$ that have connected complement. This note shows that this phenomenon can break down for non-simply connected domains Ω, even when $C \ Ω$ is compact. This answers a question of Melas and disproves a conjecture of Müller, Vlachou and Yavrian.

Il est connu que, pour un sous-domaine propre simplement connexe Ω du plan complexe et un point quelconque ζ de Ω, il y a des fonctions holomorphes sur Ω qui possèdent des séries de Taylor « universelles » autour de ζ ; c'est-à-dire tout polynôme peut être approximé, sur tout compact de $C \ Ω$ ayant un complémentaire connexe, par les sommes partielles de la série de Taylor. Cette note montre que ce résultat n'est plus vrai en général pour les domaines non-simplement connexes Ω, même lorsque $C \ Ω$ est compact. Cela répond à une question de Melas et réfute une conjecture de Müller, Vlachou et Yavrian.

Received: 2010-01-28
Accepted: 2010-03-03
Published online: 2010-03-29

DOI: 10.1016/j.crma.2010.03.003

Author's affiliations:

Stephen J. Gardiner ¹; Nikolaos Tsirivas ¹

¹ School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

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Stephen J. Gardiner; Nikolaos Tsirivas. Universal Taylor series for non-simply connected domains. Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 521-524. doi : 10.1016/j.crma.2010.03.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.03.003/

References
Cited by

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

[2] A.G. Bacharoglou Universal Taylor series on doubly connected domains, Results Math., Volume 53 (2009), pp. 9-18

[3] F. Bayart Universal Taylor series on general doubly connected domains, Bull. London Math. Soc., Volume 37 (2005), pp. 878-884

[4] C.K. Chui; M.N. Parnes Approximation by overconvergence of power series, J. Math. Anal. Appl., Volume 36 (1971), pp. 693-696

[5] G. Costakis Some remarks on universal functions and Taylor series, Math. Proc. Camb. Philos. Soc., Volume 128 (2000), pp. 157-175

[6] G. Costakis Universal Taylor series on doubly connected domains in respect to every center, J. Approx. Theory, Volume 134 (2005), pp. 1-10

[7] G. Costakis; V. Vlachou Universal Taylor series on non-simply connected domains, Analysis, Volume 26 (2006), pp. 347-363

[8] W. Gehlen Overconvergent power series and conformal maps, J. Math. Anal. Appl., Volume 198 (1996), pp. 490-505

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

[10] E. Hille Analytic Function Theory, vol. II, Ginn, Boston, 1962

[11] J.-P. Kahane Baire's category theorem and trigonometric series, J. Anal. Math., Volume 80 (2000), pp. 143-182

[12] W. Luh Universal approximation properties of overconvergent power series on open sets, Analysis, Volume 6 (1986), pp. 191-207

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

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

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

[16] J. Müller Small domains of overconvergence of power series, J. Math. Anal. Appl., Volume 172 (1993), pp. 500-507

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

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

[19] V. Nestoridis; C. Papachristodoulos Universal Taylor series on arbitrary planar domains, C. R. Math. Acad. Sci. Paris, Volume 347 (2009) no. 7–8, pp. 363-367

[20] G. Pólya Untersuchungen über Lücken und Singularitäten von Potenzreihen. II, Ann. of Math. (2), Volume 34 (1933), pp. 731-777

[21] T. Ransford Potential Theory in the Complex Plane, Cambridge Univ. Press, Cambridge, 1995

[22] N. Tsirivas, Universal Faber and Taylor series on an unbounded domain of infinite connectivity, Complex Var. Theory Appl., in press

[23] N. Tsirivas; V. Vlachou Universal Faber series with Hadamard–Ostrowski gaps, Comput. Methods Funct. Theory, Volume 10 (2010), pp. 155-165

[24] V. Vlachou A universal Taylor series in the doubly connected domain $C \ {1}$ , Complex Var. Theory Appl., Volume 47 (2002), pp. 123-129

[25] V. Vlachou Universal Taylor series on a non-simply connected domain and Hadamard–Ostrowski gaps, Complex and Harmonic Analysis, DEStech Publ., Inc., Lancaster, PA, 2007, pp. 221-229

Cited by Sources:

^☆ This research was supported by Science Foundation Ireland under Grant 09/RFP/MTH2149, and is also part of the programme of the ESF Network “Harmonic and Complex Analysis and Applications” (HCAA).

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