Universal Taylor series
Annales de l'Institut Fourier, Volume 46 (1996) no. 5, pp. 1293-1306.

We strengthen a result of Chui and Parnes and we prove that the set of universal Taylor series is a G δ -dense subset of the space of holomorphic functions defined in the open unit disc. Our result provides the answer to a question stated by S.K. Pichorides concerning the limit set of Taylor series. Moreover, we study some properties of universal Taylor series and show, in particular, that they are trigonometric series in the sense of D. Menchoff.

Nous améliorons un résultat de Chui et Parnes et nous démontrons que les séries de Taylor universelles forment un sous-espace G δ -dense de l’espace de fonctions holomorphes définies sur le disque unité ouvert. Nous utilisons ce résultat pour répondre à une question de S.K. Pichorides sur l’ensemble limite des séries de Taylor. Nous étudions aussi quelques propriétés des séries de Taylor universelles; en particulier, ce sont des séries trigonométriques universelles au sens de Menchoff.

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     title = {Universal {Taylor} series},
     journal = {Annales de l'Institut Fourier},
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     volume = {46},
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Nestoridis, Vassili. Universal Taylor series. Annales de l'Institut Fourier, Volume 46 (1996) no. 5, pp. 1293-1306. doi : 10.5802/aif.1549. https://aif.centre-mersenne.org/articles/10.5802/aif.1549/

[1] N. K. Bary, A Treatise on Trigonometric Series, Vol. I, II, Pergamon Press, 1964. | MR | Zbl

[2] C. Chui, M. N. Parnes, Approximation by overconvergence of power series, Journal of Mathematical Analysis and Applications, 36 (1971), 693-696. | MR | Zbl

[3] C. Chui, M. N. Parnes, Limit set of power series outside the circles of convergence, Pacific Journal of Mathematics, 50 (1974), 403-423. | MR | Zbl

[4] P. Dienes, The Taylor Series, Dover Pub. Inc., New York, 1957. | Zbl

[5] J.-P. Kahane, Sur la structure circulaire des ensembles de points limites des sommes partielles d'une série de Taylor, Acta Sci. Math. (Szeged), 45, n° 1-4 (1983), 247-251. | MR | Zbl

[6] E. S. Katsoprinakis, On a theorem of Marcinkiewicz and Zygmund for Taylor series, Arkiv for Matematik, 27, n° 1 (1989) 105-126. | MR | Zbl

[7] E. S. Katsoprinakis, V. Nestoridis, Partial sums of Taylor series on a circle, Ann. Inst. Fourier, 38-3 (1989), 715-736. | Numdam | MR | Zbl

[8] E. S. Katsoprinakis, Taylor series with limit points on a finite number of circles, Transactions of A.M.S., 337, n° 1 (1993), 437-450. | MR | Zbl

[9] E. S. Katsoprinakis, V. Nestoridis, An application of Kronecker's Theorem to rational functions, Math. Ann., 298 (1994), 145-166. | MR | Zbl

[10] Y. Katznelson, An Introduction to Harmonic Analysis, John Wiley & Sons Inc., New York, London, Sydney, Toronto, 1968. | MR | Zbl

[11] S. Kierst, E. Szpirajn, Sur certaines singularités des fonctions analytiques uniformes, Fundamental Mathematicae, 21 (1933), 267-294. | JFM | Zbl

[12] J. Marcinkiewicz, A. Zygmund, On the behaviour of triginometric series and power series, Transactions of A.M.S., 50 (1941), 407-453. | JFM | MR | Zbl

[13] D. Menchoff, Sur les séries Trigonométriques Universelles, Comptes Rendus (Doklady) de l'Académie des Sciences de l'URSS, Vol. XLIX, n° 2 (1945), 79-82. | MR | Zbl

[14] V. Nestoridis, Limit points of partial sums of Taylor series, Matematika, 38 (1991), 239-249. | MR | Zbl

[15] V. Nestoridis, Distribution of partial sums of the Taylor development of rational functions, Transactions of A.M.S., 346, n° 1 (1994), 283-295. | MR | Zbl

[16] V. Nestoridis, S. K. Pichorides, The circular structure of the set of limit points of partial sums of Taylor series, Séminaire d'Analyse Harmonique, Université de Paris-Sud, Mathématiques, Orsay, France (1989-1990), 71-77. | MR | Zbl

[17] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966. | MR | Zbl

[18] A. Zygmund, Trigonometric Series, second edition reprinted, Vol. I, II, Cambridge University Press, 1979.

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