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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
Article

Abstract

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.

Keywords

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 

Notes

Acknowledgments

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).

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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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