Complex Analysis and Operator Theory

, Volume 10, Issue 8, pp 1757–1774 | Cite as

Toeplitz Localization Operators: Spectral Functions Density

  • Ondrej Hutník
  • Egor A. Maximenko
  • Anna Mišková


We consider two classes of localization operators based on the Calderón and Gabor reproducing formulas and represent them in a uniform way as Toeplitz operators. We restrict our attention to the generating symbols depending on the first coordinate in the phase space. In this case, the Toeplitz localization operators (TLOs) exhibit an explicit diagonalization, i.e., there exists an isometric isomorphism that transforms all TLOs to the multiplication operators by some specific functions—we call them spectral functions. We show that these spectral functions can be written in the form of a convolution of the generating symbol of TLO with a kernel function incorporating an admissible wavelet/window. Using the Wiener’s deconvolution technique on the real line, we prove that the set of spectral functions is dense in the C\(^*\)-algebra of bounded uniformly continuous functions on the real line under the assumption that the Fourier transform of the kernel function does not vanish on the real line. This provides an explicit and independent description of the C\(^*\)-algebra generated by the set of spectral functions. The result is then applied to the case of a parametric family of wavelets related to Laguerre functions. Thereby we also provide an explicit description of the C\(^*\)-algebra generated by vertical Toeplitz operators on true poly-analytic Bergman spaces over the upper half-plane.


Toeplitz operator Localization operator Time-frequency analysis Wavelet transform Wiener’s deconvolution Meixner-Pollaczek polynomials Operator algebra Approximate invertibility 

Mathematics Subject Classification

Primary 47B35 42C40 Secondary 47G30 47L80 



The authors are grateful to the referees for many useful suggestions and corrections which helped to improve the final version of the paper. We also thank the referees for pointing out the reference [34]. The proof of Proposition 5 was found jointly with K. M. Esmeral García, and was inspired by joint works with C. Herrera Yañez and N. Vasilevski. The first author thanks H. G. Feichtinger for stimulating discussions during the follow-up workshop “Time-Frequency Analysis” held at the Erwin Schrödinger International Institute for Mathematical Physics in Vienna (ESI) in January 2014, and ESI for the hospitality and financial support provided. The authors gratefully acknowledge the financial support provided by internal university Grant VVGS-2014-182 (Slovakia), and by Project IPN-SIP 2016-0733 (Mexico).


  1. 1.
    Araaya, T.K.: The symmetric Meixner-Pollaczek polynomials. Uppsala dissertations in Mathematics, vol. 27, p. 70. Uppsala. ISBN 91-506-1681-1 (2003)Google Scholar
  2. 2.
    Bauer, W., Herrera Yañez, C., Vasilevski, N.: Eigenvalue characterization of radial operators on weighted Bergman spaces over the unit ball. Integr. Equ. Oper. Theory 78(2), 271–300 (2014)Google Scholar
  3. 3.
    Cordero, E., Gröchenig, K.: Time-frequency analysis of localization operators. J. Funct. Anal. 205(1), 107–131 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Daubechies, I.: Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inf. Theory 34(4), 605–612 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Esmeral, K., Maximenko, E.A., Vasilevski, N.L.: C\(^\ast \)-algebra generated by angular Toeplitz operators on the weighted Bergman spaces over the upper half-plane. Integr. Equ. Oper. Theory 83(3), 413–428 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Engliš, M.: Toeplitz operators and localization operators. Trans. Amer. Math. Soc. 361(2), 1039–1052 (2009)Google Scholar
  7. 7.
    Gradshteyn, I.S., Ryzhik, I.M.: Tables of integrals, series and products, 7th edn. Academic Press, Elsevier, San Diego (2007)zbMATHGoogle Scholar
  8. 8.
    Gröchenig, K.: Foundations of time-frequency analysis. Birkhäuser Boston Inc., Boston (2001)CrossRefzbMATHGoogle Scholar
  9. 9.
    Grudsky, S.M., Maximenko, E.A., Vasilevski, N.L.: Radial Toeplitz operators on the unit ball and slowly oscillating sequences. Commun. Math. Anal. 14(2), 77–94 (2013)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Grudsky, S., Quiroga-Barranco, R., Vasilevski, N.: Commutative C\(^\ast \)-algebras of Toeplitz operators and quantization on the unit disk. J. Funct. Anal. 234(1), 1–44 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Herrera Yañez, C., Hutník, O., Maximenko, E.A.: Vertical symbols, Toeplitz operators on weighted Bergman spaces over the upper half-plane and very slowly oscillating functions. Comptes Rendus Mathematique 352(2), 129–132 (2014)Google Scholar
  12. 12.
    Herrera Yañez, C., Maximenko, E.A., Vasilevski, N.L.: Vertical Toeplitz operators on the upper half-plane and very slowly oscillating functions. Integr. Equ. Oper. Theory 77(2), 149–166 (2013)Google Scholar
  13. 13.
    Hogan, J.A., Lakey, J.D.: Time-frequency and time-scale methods. Birkhäuser Boston Inc., Boston (2005)zbMATHGoogle Scholar
  14. 14.
    Howell, W.T.: Products of Laguerre polynomials. Phil. Mag: Series 7, 24(161), 396–405 (1937)Google Scholar
  15. 15.
    Huang, H.: Maximal Abelian von Neumann algebras and Toeplitz operators with separately radial symbols. Integr. Equ. Oper. Theory 64(3), 381–398 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hutník, O.: Wavelets from Laguerre polynomials and Toeplitz-type operators. Integr. Equ. Oper. Theory 71(3), 357–388 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hutník, O., Hutníková, M.: On Toeplitz localization operators. Arch. Math. 97(4), 333–344 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hutník, O., Hutníková, M.: Toeplitz operators on poly-analytic spaces via time-scale analysis. Oper. Matrices 8(4), 1107–1129 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hutníková, M., Hutník, O.: An alternative description of Gabor spaces and Gabor-Toeplitz operators. Rep. Math. Phys. 66(2), 237–250 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Inoue, H.: Expansion of Dirichlet L-function on the critical line in Meixner-Pollaczek polynomials. (2014). arXiv:1412.1220v1
  21. 21.
    Koekoek, R., Lesky, P.A., Swarttouw, R.: Hypergeometric orthogonal polynomials and their \(q\)-analogues, Springer monographs in mathematics. Springer-Verlag (2010)Google Scholar
  22. 22.
    Koornwinder, T.: Meixner-Pollaczek polynomials and the Heisenberg algebra. J. Math. Phys. 30(4), 767–769 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lang, S.: Undergraduate analysis, 2nd edn. Undergraduate texts in mathematics. Springer-Verlag, New York (2005)Google Scholar
  24. 24.
    Marcellán, F., Álvarez-Nodarse, R.: On the “Favard theorem” and its extensions. J. Comput. Appl. Math. 127(1–2), 231–254 (2001)Google Scholar
  25. 25.
    Nowak, K.: On Calderón-Toeplitz operators. Monatsh. Math. 116(1), 49–72 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pinsky, M.A.: Introduction to Fourier analysis and wavelets. Graduate studies in mathematics, vol. 201. Amer. Math. Soc., Providence (2002)Google Scholar
  27. 27.
    Ramanathan, J., Topiwala, P.: Time-frequency localization via the Weyl correspondence. SIAM J. Math. Anal. 24(5), 1378–1393 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ramírez Ortega, J., Sánchez-Nungaray, A.: Toeplitz operators with vertical symbols acting on the poly-Bergman spaces of the upper half-plane. Complex Anal. Oper. Theory 9(8), 1801–1817 (2015)Google Scholar
  29. 29.
    Reiter, H., Stegeman, D.: Classical harmonic analysis and locally compact groups, 2nd edn. Oxford University Press, Oxford (2001)zbMATHGoogle Scholar
  30. 30.
    Rochberg, R.: Toeplitz and Hankel operators, wavelets, NWO sequences, and almost diagonalization of operators. In: Arveson, W.B., Douglas, R.G. (eds.) Proc. Symp. Pure Math. 51, part I. Amer. Math. Soc., Providence, pp. 425–444 (1990)Google Scholar
  31. 31.
    Suárez, D.: The eigenvalues of limits of radial Toeplitz operators. Bull. Lond. Math. Soc. 40(4), 631–641 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Vasilevski, N.L.: Commutative algebras of Toeplitz operators on the Bergman space. Operator theory: advances and applications, vol. 185. Birkhäuser, Basel (2008)Google Scholar
  33. 33.
    Wong, M.W.: Wavelets transforms and localization operators. Operator theory: advances and applications, vol. 136. Birkhäuser, Boston (2002)Google Scholar
  34. 34.
    Xia, J.: Localization and the Toeplitz algebra on the Bergman space. J. Funct. Anal. 269(3), 781–841 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Ondrej Hutník
    • 1
  • Egor A. Maximenko
    • 2
  • Anna Mišková
    • 1
  1. 1.Faculty of Science, Institute of MathematicsPavol Jozef Šafárik University in KošiceKošiceSlovakia
  2. 2.Escuela Superior de Física y MatemáticasInstituto Politécnico NacionalCiudad de MéxicoMexico

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