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A version of Fabry's theorem for power series with regularly varying coefficients

Author: Alexandre Eremenko
Journal: Proc. Amer. Math. Soc. 136 (2008), 4389-4394
MSC (2000): Primary 30B10, 30B40
Published electronically: July 24, 2008
MathSciNet review: 2431054
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Abstract: For real power series whose non-zero coefficients satisfy $ \vert a_m\vert^{1/m} \to~1$, we prove a stronger version of Fabry's theorem relating the frequency of sign changes in the coefficients and analytic continuation of the sum of the power series.

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  • 1. N. U. Arakelyan and V. A. Martirosyan, The location of singularities of power series on the boundary of the disk of convergence, Izvestia Akademii Nauk Armyanskoi SSR Ser. Mat. vol. 22, No. 1 (1987) 3-21, 102 (Russian); English translation: Soviet Journal of Contemporary Mathematical Analysis, 22 (1987) 1-19. MR 898105 (88k:30003)
  • 2. N. Arakelyan, W. Luh and J. Müller, On the localization of singularities of lacunar power series, Complex Variables and Elliptic Equations, 52 (2007) 561-573. MR 2340942 (2008f:30005)
  • 3. L. Bieberbach, Analytische Fortsetzung, Springer, Berlin, 1955. MR 0068621 (16:913a)
  • 4. A. Eremenko, Densities in Fabry's theorem, preprint, arXiv:0709.2360.
  • 5. E. Fabry, Sur les séries de Taylor qui ont une infinité de points singuliers, Acta Math., 22 (1899) 65-87. MR 1554901
  • 6. B. Fuglede, Some properties of the Riesz charge associated with a $ \delta$-subharmonic function. Potential Anal., 1 (1992) 355-371. MR 1245891 (94m:31003)
  • 7. A. F. Grishin, Sets of regular growth of entire functions I, Teor. Funktsii Funktsional. Anal. i Prilozhen. No. 40 (1983) 36-47 (Russian). MR 738442 (85j:30057)
  • 8. L. Hörmander, The analysis of linear partial differential operators, vol. I, Springer, Berlin, 1983. MR 0717035 (85g:35002a)
  • 9. P. Koosis, The Logarithmic Integral, vol. II, Cambridge Univ. Press, Cambridge, 1992. MR 1195788 (94i:30027)
  • 10. B. Ya. Levin, Distribution of zeros of entire functions, AMS, Providence, RI, 1980. MR 589888 (81k:30011)
  • 11. G. Pólya, Über gewisse notwendige Determinantenkriterien fur die Fortsetzbarkeit einer Potenzreihe, Math. Ann., 99 (1928) 687-706. MR 1512473
  • 12. G. Pólya, Untersuchungen über Lücken und Singularitäten von Potenzreihen, Math. Z., 29 (1929) 549-640. MR 1545027
  • 13. C. de la Valle-Poussin, Potentiel et problème généralisé de Dirichlet, Math. Gazette, 22 (1938) 17-36.

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Additional Information

Alexandre Eremenko
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Received by editor(s): November 19, 2007
Published electronically: July 24, 2008
Additional Notes: The author was supported by NSF grant DMS-0555279.
Communicated by: Mario Bonk
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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