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Localization of Zeros in Cauchy–de Branges Spaces

  • Evgeny AbakumovEmail author
  • Anton Baranov
  • Yurii Belov
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

We study the class of discrete measures in the complex plane with the following property: up to a finite number, all zeros of any Cauchy transform of the measure (with 2-data) are localized near the support of the measure. We find several equivalent forms of this property and prove that the parts of the support attracting zeros of Cauchy transforms are ordered by inclusion modulo finite sets.

Keywords

Cauchy transforms de Branges spaces Distribution of zeros of entire functions Polynomial approximation 

1991 Mathematics Subject Classification

30D10 30D15 46E22 41A30 34B20 

References

  1. 1.
    E. Abakumov, A. Baranov, Y. Belov, Localization of zeros for Cauchy transforms. Int. Math. Res. Notices 2015(15), 6699–6733 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    E. Abakumov, A. Baranov, Y. Belov, Krein-type theorems and ordered structure of Cauchy–de Branges spaces. J. Funct. Anal. (2019, to appear). arXiv:1802.03385Google Scholar
  3. 3.
    N.I. Akhiezer, On the weighted approximation of continuous functions by polynomials on the real axis. Uspekhi Mat. Nauk 11 (1956), 3–43; AMS Transl. (Ser. 2) 22, 95–137 (1962)Google Scholar
  4. 4.
    A. Baranov, Spectral theory of rank one perturbations of normal compact operators. Algebra Anal. 30(5), 1–56 (2018)MathSciNetGoogle Scholar
  5. 5.
    A.D. Baranov, D.V. Yakubovich, Completeness and spectral synthesis of nonselfadjoint one-dimensional perturbations of selfadjoint operators. Adv. Math. 302, 740–798 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    A.D. Baranov, D.V. Yakubovich, Completeness of rank one perturbations of normal operators with lacunary spectrum. J. Spectral Theory 8(1), 1–32 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Baranov, Y. Belov, A. Borichev, Summability properties of Gabor expansions. J. Funct. Anal. 274(9), 2532–2552 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Y. Belov, T. Mengestie, K. Seip, Discrete Hilbert transforms on sparse sequences. Proc. Lond. Math. Soc. 103(3), 73–105 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Borichev, M. Sodin, Weighted polynomial approximation and the Hamburger moment problem, in Complex Analysis and Differential Equations, Proceedings of the Marcus Wallenberg Symposium in Honor of Matts Essén (Uppsala University, Uppsala, 1998)Google Scholar
  10. 10.
    A. Borichev, M. Sodin, The Hamburger moment problem and weighted polynomial approximation on discrete subsets of the real line. J. Anal. Math. 76, 219–264 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    J. Clunie, A. Eremenko, J. Rossi, On equilibrium points of logarithmic and Newtonian potentials. J. Lond. Math. Soc. 47(2), 309–320 (1993)MathSciNetCrossRefGoogle Scholar
  12. 12.
    L. de Branges, Hilbert Spaces of Entire Functions (Prentice–Hall, Englewood Cliffs, 1968)Google Scholar
  13. 13.
    A. Eremenko, J. Langley, J. Rossi, On the zeros of meromorphic functions of the form \(f(z)=\sum _{k=1}^\infty a_k/(z-z_k)\). J. Anal. Math. 62, 271–286 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    V. Havin, B. Jöricke, The Uncertainty Principle in Harmonic Analysis (Springer, Berlin, 1994)CrossRefGoogle Scholar
  15. 15.
    P. Koosis, The Logarithmic Integral. I (Cambridge University Press, Cambridge, 1988)Google Scholar
  16. 16.
    J.K. Langley, J. Rossi, Meromorphic functions of the form \(f(z)=\sum _{n=1}^\infty a_n/(z-z_n)\). Rev. Mat. Iberoamericana 20(1), 285–314 (2004)Google Scholar
  17. 17.
    S. Mergelyan, Weighted approximation by polynomials. Uspekhi Mat. Nauk 11, 107–152 (1956); AMS Transl. 10, 59–106 (1958)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University Paris-Est, LAMA (UMR 8050), UPEM, UPEC, CNRS,Marne-la-ValléeFrance
  2. 2.Department of Mathematics and MechanicsSt. Petersburg State UniversitySt. PetersburgRussia
  3. 3.National Research University Higher School of EconomicsSt. PetersburgRussia
  4. 4.St. Petersburg State UniversitySt. PetersburgRussia

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