Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. N. U. Arakelyan and V. A. Martirosyan, The location of singularities of power series on the boundary of the disk of convergence, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 22 (1987), no. 1, 3–21, 102 (Russian, with English and Armenian summaries). MR 898105
  • 2. Norair Arakelian, Wolfgang Luh, and Jürgen Müller, On the localization of singularities of lacunar power series, Complex Var. Elliptic Equ. 52 (2007), no. 7, 561–573. MR 2340942, 10.1080/17476930701246396
  • 3. Ludwig Bieberbach, Analytische Fortsetzung, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 3, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955 (German). MR 0068621
  • 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), no. 1, 65–87 (French). MR 1554901, 10.1007/BF02417871
  • 6. Bent Fuglede, Some properties of the Riesz charge associated with a ∂-subharmonic function, Potential Anal. 1 (1992), no. 4, 355–371. MR 1245891, 10.1007/BF00301788
  • 7. A. F. Grishin, Sets of regular growth of entire functions. I, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 40 (1983), 36–47 (Russian). MR 738442
  • 8. Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035
  • 9. Paul Koosis, The logarithmic integral. II, Cambridge Studies in Advanced Mathematics, vol. 21, Cambridge University Press, Cambridge, 1992. MR 1195788
  • 10. B. Ja. Levin, Distribution of zeros of entire functions, Revised edition, Translations of Mathematical Monographs, vol. 5, American Mathematical Society, Providence, R.I., 1980. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. MR 589888
  • 11. G. Pólya, Über gewisse notwendige Determinantenkriterien für die Fortsetzbarkeit einer Potenzreihe, Math. Ann. 99 (1928), no. 1, 687–706 (German). MR 1512473, 10.1007/BF01459120
  • 12. G. Pólya, Untersuchungen über Lücken und Singularitäten von Potenzreihen, Math. Z. 29 (1929), no. 1, 549–640 (German). MR 1545027, 10.1007/BF01180553
  • 13. C. de la Valle-Poussin, Potentiel et problème généralisé de Dirichlet, Math. Gazette, 22 (1938) 17-36.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 30B10, 30B40

Retrieve articles in all journals with MSC (2000): 30B10, 30B40


Additional Information

Alexandre Eremenko
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: eremenko@math.purdue.edu

DOI: https://doi.org/10.1090/S0002-9939-08-09652-4
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.