|
A version of Fabry's theorem for power series with regularly varying coefficients
Author(s):
Alexandre
Eremenko
Journal:
Proc. Amer. Math. Soc.
136
(2008),
4389-4394.
MSC (2000):
Primary 30B10, 30B40
Posted:
July 24, 2008
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
For real power series whose non-zero coefficients satisfy  , 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:
-
- 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
 -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.
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:
10.1090/S0002-9939-08-09652-4
PII:
S 0002-9939(08)09652-4
Received by editor(s):
November 19, 2007
Posted:
July 24, 2008
Additional Notes:
The author was supported by NSF grant DMS-0555279.
Communicated by:
Mario Bonk
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org