Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Regularly distributed subsets in the complex plane


Authors: A. I. Abdulnagimov and A. S. Krivosheyev
Translated by: N. N. Osipov
Original publication: Algebra i Analiz, tom 28 (2016), nomer 4.
Journal: St. Petersburg Math. J. 28 (2017), 433-464
MSC (2010): Primary 30D20
DOI: https://doi.org/10.1090/spmj/1458
Published electronically: May 4, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Certain conditions are studied under which there exists a regularly distributed set that is a part of a given sequence of complex numbers and, moreover, contains a given subsequence of that sequence. On this basis, splitting of entire functions and their asymptotic behavior are investigated. The results are also applied to problems concerning the completeness of systems of exponential monomials in convex domains and the representation of functions analytic on compact convex sets, as well as to the fundamental principle problem for invariant subspaces of functions.


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

  • 1. B. Ya. Levin, Distribution of zeros on entire functions, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956. MR 0087740 (19,402c); English transl., Transl. Math. Monogr., vol. 5, Amer. Math. Soc., Providence, RI, 1980. MR 0156975 (28:217)
  • 2. A. F. Leont'ev, Exponential series, Nauka, Moscow, 1976. (Russian) MR 0584943
  • 3. G. Polya, Untersuchungen uber Lücken und Singularitaten von Potenzreihen, Math. Z. 29 (1929), no. 1, 549-640. MR 1545027
  • 4. A. A. Kondratyuk, Entire functions with positive zeros that have a finite maximal density, Teor. Funkcii Funkcional. Anal. i Prilozhen. (Kharkov) 7 (1968), 37-52. (Russian) MR 0277717
  • 5. G. L. Luntz, A certan theorem that is connected with the grawth of entire functions of entire order, Izv. Akad. Nauk Armjan. SSR. Ser. Mat. 5 (1970), no. 4, 358-370. (Russian) MR 0288264
  • 6. G. N. Shilova, A theorem on the divisors of finite-order entire functions, Mat. Zametki 48 (1990), no. 2, 128-136; English transl., Math. Notes 48 (1990), no. 1-2, 799-804. MR 1076943
  • 7. A. I. Abdulagimov and A. S. Krovosheyev, Properly distributed subsequence on the line, Ufim. Mat. Zh. 7 (2015), no. 1, 3-12; English transl., Ufa Math. J. 7 (2015), no. 1, 3-12. MR 3430737
  • 8. M. L. Cartwright, On integral functions of integral order, Proc. London Math. Soc. 33 (1931), no. 1, 209-224. MR 1576826
  • 9. A. S. Krivosheev, The fundamental principle for invariant subspaces in convex domains, Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 2, 71-136; English transl., Izv. Math. 68 (2004), no. 2, 291-353. MR 2058001
  • 10. O. A. Krivosheeva and A. S. Krivosheev, Criterion for the fudamental principle to hold for invariant subspaces in bounded convex domains in the complex plane, Functsional. Anal. i Prilozhen. 46 (2012), no. 4, 14-30; English transl., Funct. Anal. Appl. 46 (2012), no. 4, 249-261. MR 3075093
  • 11. -, A closedness of set of Dirichlet series sums, Ufim. Mat. Zh. 5 (2013), no. 3, 96-120; English transl., Ufa Math. J. 5 (2013), no. 3, 94-117. MR 3430790
  • 12. A. A. Kondratyuk, Entire functions with finite maximal density of zeros, Teor. Funkcii Funkcional. Anal. i Prilozhen. (Kharkov) 10 (1970), 57-70. (Russian) MR 0298000
  • 13. A. F. Leont'ev, Entire functions. Series of exponentials, Nauka, Moscow, 1983. (Russian) MR 0753827 (86j:30005)
  • 14. V. V. Napalkov, Convolution equations in multidimensional spaces, Nauka, Moscow, 1982. (Russian) MR 0678923
  • 15. O. A. Krivosheeva, Singular points of the sum of series of exponential monomials on the boundary of the convergence domain, Algebra i Analiz 23 (2011), no. 2, 162-205; English transl., St. Peterburg Math. J. 23 (2012), no. 2, 321-350. MR 2841675
  • 16. O. A. Krivosheeva and A. S. Krovosheev, Singular points of the sum of a Dirichlet series on the convergence line, Functsional. Anal. i Prilozhen. 49 (2015), no. 2. 54-69; English transl., Funct. Anal. Appl. 49 (2015), no. 2, 122-134. MR 3374903
  • 17. O. A. Krivosheeva, The convergence domain for series of exponential monomials, Ufim. Mat. Zh. 3 (2011), no. 2, 43-56; English transl., Ufa Math. J. 3 (2011), no. 2, 42-55. MR 3428992
  • 18. I. F. Krasičkov-Ternovskiĭ, Invariant subspaces of analytic functions. II. Spectral synthesis on convex domains, Mat. Sb. 88 (1972), no. 1, 3-30. (Russian) MR 0422636
  • 19. A. A. Gol'dberg, B. Ya. Levin, and I. V. Ostrovskiĭ, Entire and meromorphic functions. Itogi Nauki i Tekhniki, Sovremen. Probl. Math., vol. 256, VINITI, Moscow, 1991, pp. 5-186. MR 1155417

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 30D20

Retrieve articles in all journals with MSC (2010): 30D20


Additional Information

A. I. Abdulnagimov
Affiliation: Institute of mathematics with computer center, Ufa scientific center Russian Academy of Sciences, Chernyshevskiĭ str. 112, 450048 Ufa, Russia
Email: buffonishe@mail.ru

A. S. Krivosheyev
Affiliation: Bashkir State University, Zaki Validi str. 32, 450076 Ufa, Russia
Email: kriolesya2006@yandex.ru

DOI: https://doi.org/10.1090/spmj/1458
Keywords: Regularly distributed set, series of exponential functions, entire function, convex domain
Received by editor(s): June 25, 2015
Published electronically: May 4, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society