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Universal Taylor series. (English) Zbl 0865.30001
Summary: We strengthen a result of Chui and Parnes and we prove that the set of universal Taylor series is a \(G_\delta\)-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.

MSC:
30B50 Dirichlet series, exponential series and other series in one complex variable
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
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References:
[1] N. K. BARY, A treatise on trigonometric series, Vol. I, II, Pergamon Press, 1964. · Zbl 0129.28002
[2] C. CHUI, M. N. PARNES, Approximation by overconvergence of power series, Journal of Mathematical Analysis and Applications, 36 (1971), 693-696. · Zbl 0224.30006
[3] C. CHUI, M. N. PARNES, Limit set of power series outside the circles of convergence, Pacific Journal of Mathematics, 50 (1974), 403-423. · Zbl 0255.30001
[4] P. DIENES, The Taylor series, Dover Pub. Inc., New York, 1957. · Zbl 0078.05901
[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. · Zbl 0528.30004
[6] E. S. KATSOPRINAKIS, On a theorem of Marcinkiewicz and Zygmund for Taylor series, Arkiv for Matematik, 27, n° 1 (1989) 105-126. · Zbl 0676.42004
[7] E. S. KATSOPRINAKIS, V. NESTORIDIS, Partial sums of Taylor series on a circle, Ann. Inst. Fourier, 38-3 (1989), 715-736. · Zbl 0701.30003
[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. · Zbl 0774.30006
[9] E. S. KATSOPRINAKIS, V. NESTORIDIS, An application of Kronecker’s theorem to rational functions, Math. Ann., 298 (1994), 145-166. · Zbl 0787.30001
[10] Y. KATZNELSON, An introduction to harmonic analysis, John Wiley & Sons Inc., New York, London, Sydney, Toronto, 1968. · Zbl 0169.17902
[11] S. KIERST, E. SZPIRAJN, Sur certaines singularités des fonctions analytiques uniformes, Fundamental Mathematicae, 21 (1933), 267-294. · JFM 59.0328.02
[12] J. MARCINKIEWICZ, A. ZYGMUND, On the behaviour of triginometric series and power series, Transactions of A.M.S., 50 (1941), 407-453. · JFM 67.0225.02
[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. · Zbl 0060.18504
[14] V. NESTORIDIS, Limit points of partial sums of Taylor series, Matematika, 38 (1991), 239-249. · Zbl 0759.30003
[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. · Zbl 0818.30001
[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. · Zbl 0724.41025
[17] W. RUDIN, Real and complex analysis, McGraw-Hill, New York, 1966. · Zbl 0142.01701
[18] A. ZYGMUND, Trigonometric series, second edition reprinted, Vol. I, II, Cambridge University Press, 1979. · JFM 58.0296.09
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