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Toeplitz Operators Associated with the Deformed Windowed Fourier Transform

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Abstract

This article deals with the study of Toeplitz operators associated with the deformed windowed Fourier transform. The primary aim is to study the Toeplitz operators, particular their trace class properties and demonstrate that they belong to the Schatten-von Neumann class of operators. Nevertheless, significant attention is also devoted to a spectral analysis by examining the eigenvalues and eigenfunctions of the time-frequency Toeplitz operator. Towards the culmination, the spectrograms associated with the deformed windowed Fourier transform are also studied in detail.

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References

  1. Abreu, D., Gröchenig, K., Romero, J.L.: On accumulated spectrograms. Trans. Am. Math. Soc. 368, 3629–3649 (2016)

    Article  MathSciNet  Google Scholar 

  2. Ben-Saïd, S., Kobayashi, T., Ørsted, B.: Laguerre semigroup and Dunkl operators. Compos. Math. 148(04), 1265–1336 (2012)

    Article  MathSciNet  Google Scholar 

  3. Ben-Saïd, S., Deleaval, L.: Translation operator and maximal function for the \((k,1)\)-generalized Fourier transform. J. Funct. Anal. 279(8), 108706 (2020)

    Article  MathSciNet  Google Scholar 

  4. Boubatra, M.A., Negzaoui, S., Sifi, M.: A new product formula involving Bessel functions. ar**v:2011.08104

  5. Cohen, L.: Time-frequency distributions-a review. Proc. IEEE 77, 941–981 (1989)

    Article  Google Scholar 

  6. Constales, D., De-Bie, H., Lian, P.: Explicit formulas for the Dunkl dihedral kernel and the \((\kappa, a)\)-generalized Fourier kernel. J. Math. Anal. Appl. 460(2), 900–926 (2018)

    Article  MathSciNet  Google Scholar 

  7. Cordero, E., Gröchenig, K.: Time-frequency analysis of localization operators. J. Funct. Anal. 205, 107–131 (2003)

    Article  MathSciNet  Google Scholar 

  8. Czaja, W., Gigante, G.: Continuous Gabor transform for strong hypergroups. J. Fourier Anal. Appl. 9, 321–339 (2003)

    Article  MathSciNet  Google Scholar 

  9. Daubechies, I.: Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inf. Theory. 34(4), 605–612 (1998)

    Article  MathSciNet  Google Scholar 

  10. Debnath, L., Shah, F.A.: Wavelet Transforms and Their Applications, Birkhäuser, (2015)

  11. Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 1(311), 167–183 (1989)

    Article  MathSciNet  Google Scholar 

  12. Dunkl, C.F.: Hankel transforms associated to finite reflection groups. In: Proceedings of the special session on hyper-geometric functions on domains of positivity, Jack polynomials and applications. (Tampa, FL, 1991), Contemp. Math. 138, 123–138 (1992)

  13. Gorbachev, D., Ivanov, V., Tikhonov, S.: Pitt’s inequalities and uncertainty principle for generalized Fourier transform. Int. Math. Res. Notices 23, 7179–7200 (2016)

    Article  MathSciNet  Google Scholar 

  14. Gröchenig, K.: Foundations of time-frequency analysis. Birkhäuser Boston Inc, Boston, MA (2001)

    Book  Google Scholar 

  15. Howe, R.: The oscillator semigroup: The mathematical heritage of Hermann Weyl (Durham, NC, 1987), 61-132, Proc. Sympos. Pure Math. 48, Am. Math. Soc. Providence, RI, (1988)

  16. Johansen, T.R.: Weighted inequalities and uncertainty principles for the \((k, a)\)-generalized Fourier transform. Int. J. Math. 27(3), 1650019 (2016)

    Article  MathSciNet  Google Scholar 

  17. Liu, L.: A trace class operator inequality. J. Math. Anal. Appl. 328, 1484–1486 (2007)

    Article  MathSciNet  Google Scholar 

  18. Mejjaoli, H.: Spectral theorems associated with the \((k, a)\)-generalized wavelet multipliers. J. Pseudo-Differ. Oper. Appl. 9, 735–762 (2018)

    Article  MathSciNet  Google Scholar 

  19. Mejjaoli, H.: \((k, a)\)-generalized wavelet transform and applications. J. Pseudo-Differ. Oper. Appl. 11, 55–92 (2020)

    Article  MathSciNet  Google Scholar 

  20. Mejjaoli, H.: New uncertainty principles for the \((k,a)\)-generalized wavelet transform. Revista de la Unión Matemática Argentina. (2022). https://doi.org/10.33044/revuma.2051

  21. Mejjaoli, H.: Deformed Gabor transform and applications. Int. J. Open Problems Compt. Math. 14(2), 86–123 (2021)

    Google Scholar 

  22. Mejjaoli, H.: Deformed Stockwell transform and applications on the reproducing kernel theory. Int. J. Reprod. Kernels. (to appear) (2021)

  23. Riesz, F., Sz Nagy, B.: Functional Analysis. Frederick Ungar Publishing Co., New York (1995)

    MATH  Google Scholar 

  24. Stein, E.M.: Interpolation of linear operators. Trans. Amer. Math. Soc. 83, 482–492 (1956)

    Article  MathSciNet  Google Scholar 

  25. Trimèche, K.: Generalized Wavelets and Hypergroups. Gordon and Breach Science Publishers, London (1997)

    MATH  Google Scholar 

  26. Wong, M.W.: Wavelet Transforms and Localization Operators. Springer, Berlin (2002)

    Book  Google Scholar 

Download references

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Authors and Affiliations

  1. College of Sciences, Department of Mathematics, Taibah University, PO BOX 30002, Al Madinah AL Munawarah, Saudi Arabia

    Hatem Mejjaoli

  2. Department of Mathematics, University of Kashmir, South Campus, Anantnag, Jammu and Kashmir, 192101, India

    Firdous A. Shah

  3. Department of Mathematics, Faculty Sciences of Gabes, Gabes University, Omar Ibn Khattab Street, 6029, Gabes, Tunisia

    Nadia Sraieb

Authors
  1. Hatem Mejjaoli
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  2. Firdous A. Shah
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  3. Nadia Sraieb
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Correspondence to Firdous A. Shah.

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The authors declare that they have no conflicts of interest.

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Communicated by Palle Jorgenson.

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This article is part of the topical collection “Harmonic Analysis and Operator Theory” edited by H. Turgay Kaptanoglu, Andreas Seeger, Franz Luef and Serap Oztop.

Hatem Mejjaoli thanks professors K. Trimèche and M.W. Wong for their helps.

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Mejjaoli, H., Shah, F.A. & Sraieb, N. Toeplitz Operators Associated with the Deformed Windowed Fourier Transform. Complex Anal. Oper. Theory 16, 31 (2022). https://doi.org/10.1007/s11785-022-01210-1

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