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Beurling-Malliavin theory for Toeplitz kernels

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Abstract

We consider the family of Toeplitz operators \(T_{J\bar{S}^{a}}\) acting in the Hardy space H 2 in the upper halfplane; J and S are given meromorphic inner functions, and a is a real parameter. In the case where the argument of S has a power law type behavior on the real line, we compute the critical value

$$c(J,S)=\inf\left\{a:\mathop{\mathrm{ker}}T_{J\bar{S}^{a}}\ne0\right\}.$$

The formula for c(J,S) generalizes the Beurling-Malliavin theorem on the radius of completeness for a system of exponentials.

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References

  1. Baranov, A.: Completeness and Riesz bases of reproducing kernels in model subspaces, Int. Math. Res. Not., 2006, Article ID 81530, 34 pp.

  2. Beurling, A., Malliavin, P.: On Fourier transforms of measures with compact support. Acta Math. 107, 291–302 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  3. Beurling, A., Malliavin, P.: On the closure of characters and the zeros of entire functions. Acta Math. 118, 79–93 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bruna, J., Olevskii, A., Ulanovskii, A.: Completeness in L 1(ℝ) of discrete translates and related questions for quasi-analytic classes. Rev. Mat. Iberoam. 22, 1–16 (2005)

    Article  MathSciNet  Google Scholar 

  5. Calderón, A., Zygmund, A.: On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  6. Coifman, R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. AMS 83, 569–645 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  7. De Branges, L.: Hilbert Spaces of Entire Functions. Prentice-Hall, Englewood Cliffs (1968)

    MATH  Google Scholar 

  8. Garnett, J.: Bounded Analytic Functions. Academic Press, San Diego (1981)

    MATH  Google Scholar 

  9. Gohberg, I., Krein, M.: Theory and Applications of Volterra Operators in Hilbert Space. AMS, Providence (1970)

    MATH  Google Scholar 

  10. Havin, V., Jöricke, B.: The Uncertainty Principle in Harmonic Analysis. Springer, Berlin (1994)

    MATH  Google Scholar 

  11. Havin, V., Mashreghi, J.: Admissible majorants for model subspaces of H 2; I. Slow winding of the generating inner function, II. Fast winding of the generating inner function. Canad. J. Math. 55, 1231–1301 (2003)

    MATH  MathSciNet  Google Scholar 

  12. Higgins, J.: Completeness and Basis Properties of Sets of Special Functions. Cambridge University Press, Cambridge (1977)

    Book  MATH  Google Scholar 

  13. Hruschev, S., Nikolskii, N., Pavlov, B.: Unconditional Bases of Exponentials and of Reproducing Kernels. Lect. Notes Math., vol. 864, pp. 214–335. Springer, Berlin (1981)

    Google Scholar 

  14. Hunt, R., Muckenhoupt, B., Wheeden, R.: Weighted norm inequalities for the conjugate functions and Hilbert transform. Trans. Am. Math. Soc. 176, 227–251 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kahane, J.-P.: Travaux de Beurling et Malliavin. Seminaire Bourbaki. Exposés 223 á 228 (1962)

  16. Khabibullin, B.: Completeness of Exponential Systems and Uniqueness Sets. Bashkir State University Press, Ufa (2006)

    Google Scholar 

  17. Koosis, P.: Introduction to H p Spaces. Cambridge University Press, Cambridge (1980)

    MATH  Google Scholar 

  18. Koosis, P.: The Logarithmic Integral, vols. I & II. Cambridge University Press, Cambridge (1988)

    Book  Google Scholar 

  19. Koosis, P.: Lecons sur le Theorem de Beurling et Malliavin. Les Publications CRM, Montreal (1996)

    MATH  Google Scholar 

  20. Koosis, P.: Kargaev’s proof of Beurling’s lemma. Unpublished manuscript

  21. Levin, B.: Distribution of Zeros of Entire Functions. Am. Math. Soc., Providence (1980)

    Google Scholar 

  22. Levinson, N.: Gap and Density Theorems. Am. Math. Soc., Providence (1940)

    Google Scholar 

  23. Lyubarskii, J., Seip, K.: Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt’s (A p ) condition. Rev. Mat. Iberoam. 13, 361–376 (1997)

    MATH  MathSciNet  Google Scholar 

  24. Makarov, N., Poltoratski, A.: Meromorphic inner functions, Toeplitz kernels, and the uncertainty principle. In: Perspectives in Analysis, pp. 185–252. Springer, Berlin (2005)

    Chapter  Google Scholar 

  25. Mashreghi, J., Nazarov, F., Havin, V.: Beurling-Malliavin multiplier theorem: the seventh proof. St. Petersbg. Math. J. 17, 699–744 (2006)

    Article  MATH  Google Scholar 

  26. Nazarov, F.: The Beurling lemma via the Bellman function. Unpublished manuscript

  27. Nikolskii, N.K.: Bases of exponentials and the values of reproducing kernels. Dokl. Acad. Nauk SSSR 252, 1316–1320 (1980)

    Google Scholar 

  28. Nikolskii, N.: Operators, Functions, and Systems: An Easy Reading, vols. I & II. Am. Math. Soc., Providence (2002)

    Google Scholar 

  29. Ortega-Cedrá, J., Seip, K.: Fourier frames. Ann. Math. 155, 789–806 (2002)

    Article  Google Scholar 

  30. Paley, R., Wiener, N.: Fourier Transform in the Complex Domains. Am. Math. Soc., Providence (1934)

    Google Scholar 

  31. Pavlov, B.: The basis property of a system of exponentials and the condition of Muckenhoupt. Dokl. Acad. Nauk SSSR 247, 37–40 (1979)

    Google Scholar 

  32. Redheffer, R.: Completeness of sets of complex exponentials. Adv. Math. 24, 1–62 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  33. Schwartz, L.: Études des Sommes d’Exponentielles Réelles. Hermann, Paris (1943)

    Google Scholar 

  34. Seip, K.: Interpolation and Sampling in Spaces of Analytic Functions. Am. Math. Soc., Providence (2004)

    MATH  Google Scholar 

  35. Treil, S., Volberg, A.: Embedding theorems for invariant subspaces of inverse shift. In: Proceedings of LOMI Seminars, vol. 149, pp. 38–51 (1986)

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Authors and Affiliations

  1. California Institute of Technology, Department of Mathematics, Pasadena, CA, 91125, USA

    N. Makarov

  2. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    A. Poltoratski

Authors
  1. N. Makarov
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  2. A. Poltoratski
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Correspondence to A. Poltoratski.

Additional information

The first author is supported by N.S.F. Grant No. 0201893.

The second author is supported by N.S.F. Grant No. 0500852.

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Makarov, N., Poltoratski, A. Beurling-Malliavin theory for Toeplitz kernels. Invent. math. 180, 443–480 (2010). https://doi.org/10.1007/s00222-010-0234-2

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