Journal of Fourier Analysis and Applications

, Volume 23, Issue 4, pp 877–885 | Cite as

Gap Problem for Separated Sequences and Beurling–Malliavin Theorem

  • Anton Baranov
  • Yurii Belov
  • Alexander Ulanovskii


We show that the Gap Theorem for separated sequences by M. Mitkovski and A. Poltoratski can be deduced directly from the classical Beurling–Malliavin formula for the radius of completeness.


Gap problem Beurling–Malliavin density Exponential systems Completeness 

Mathematics Subject Classification

42A38 42A65 



This work was supported by Russian Science Foundation Grant 14-21-00035.


  1. 1.
    Beurling, A., Malliavin, P.: On the closure of characters and the zeros of entire functions. Acta Math. 118, 79–93 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Havin, V., Jöricke, B.: The Uncertainty Principle in Harmonic Analysis. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar
  3. 3.
    Koosis, P.: The Logarithmic Integral. II. Cambridge University Press, Cambridge (1992)CrossRefzbMATHGoogle Scholar
  4. 4.
    Krasichkov-Ternovskii, I.F.: An interpretation of the Beurling–Malliavin theorem on the radius of completeness. Mat. Sb. 180(3), 397–423 (1989)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Mitkovski, M., Poltoratski, A.: Pólya sequences, Toeplitz kernels and gap theorems. Adv. Math. 224, 1057–1070 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Olevskii, A., Ulanovskii, A.: Functions with Disconnected Spectrum: Sampling, Interpolation, Translates. University Lecture Series, vol. 65. AMS (2016)Google Scholar
  7. 7.
    Olevskii, A., Ulanovskii, A.: On the duality between sampling and interpolation. Anal. Math. 42(1), 43–53 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Poltoratski, A.: Spectral gaps for sets and measures. Acta Math. 208(1), 151–209 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Redheffer, R.: Two consequences of the Beurling–Malliavin theory. Proc. Am. Math. Soc. 36, 116–122 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Redheffer, R.: Completeness of sets of complex exponentials. Adv. Math. 24(1), 1–62 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ya, B.: Levin, Lectures on Entire Functions, Translations of Mathematical Monographs, 150. American Mathematical Society, Providence, RI (1996)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Anton Baranov
    • 1
  • Yurii Belov
    • 2
  • Alexander Ulanovskii
    • 3
  1. 1.Department of Mathematics and MechanicsSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Chebyshev LaboratorySt. Petersburg State UniversitySt. PetersburgRussia
  3. 3.Stavanger UniversityStavangerNorway

Personalised recommendations