Boundedness and Compactness of the Spherical Mean Two-Wavelet Localization Operators

Abstract

In this paper, we prove the boundedness and compactness of localization operators associated with spherical mean wavelet transforms, which depend on a symbol and two spherical mean wavelets on \(L^{p}(d\nu )\), \(1 \le p \le \infty \).

This is a preview of subscription content, access via your institution.

References

  1. Baccar, C., Omri, S., Rachdi, L.T.: Fock spaces connected with spherical mean operator and associated operators. Mediterr. J. Math. 6(1), 1–25 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  2. Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, New York (1988)

    Google Scholar 

  3. Boggiatto, P., Wong, M.W.: Two-wavelet localization operators on \(L^{p}(\mathbb{R}^{d})\) for the Weyl-Heisenberg group. Integr. Equ. Oper. Theory 49, 1–10 (2004)

    Article  Google Scholar 

  4. Calderon, J.P.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24, 113–190 (1964)

    MathSciNet  Article  MATH  Google Scholar 

  5. Cordero, E., Gröchenig, K.: Time-frequency analysis of localization operators. J. Funct. Anal. 205(1), 107–131 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  6. Daubechies, I.: Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inf. Theory 34(4), 605–612 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  7. Daubechies, I., Paul, T.: Time-frequency localization operators a geometric phase space approach: II. The use of dilations. Inverse Probl. 4(3), 661–680 (1988)

    Article  MATH  Google Scholar 

  8. Daubechies, I.: The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory 36(5), 961–1005 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  9. Daubechies, I.: Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61. SIAM, Philadelphia (1992)

    Google Scholar 

  10. Donoho, D.L., Stark, P.B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49, 906–931 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  11. Du, J., Wong, M.W.: Traces of wavelet multipliers. C. R. Math. Rep. Acad. Sci. Can. 23, 148–152 (2001)

    MathSciNet  MATH  Google Scholar 

  12. De Mari, F., Feichtinger, H., Nowak, K.: Uniform eigenvalue estimates for time-frequency localization operators. J. Lond. Math. Soc. 65(3), 720–732 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  13. De Mari, F., Nowak, K.: Localization type Berezin—Toeplitz operators on bounded symmetric domains. J. Geom. Anal. 12(1), 9–27 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  14. Gröchenig, K.: Foundations of Time–Frequency Analysis. Springer, Berlin (2001)

    Google Scholar 

  15. Fawcett, J.A.: Inversion of N-dimensional spherical means. SIAM. J. Appl. Math. 45, 336–341 (1983)

    MathSciNet  Article  MATH  Google Scholar 

  16. Folland, G.B.: Introduction to Partial Differential Equations, 2nd edn. Princeton University Press, Princeton (1995)

    Google Scholar 

  17. He, Z., Wong, M.W.: Wavelet multipliers and signals. J. Austral. Math. Soc. Ser. B 40, 437–446 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  18. Helesten, H., Andersson, L.E.: An inverse method for the processing of synthetic aperture radar data. Inv. Prob. 3, 111–124 (1987)

    MathSciNet  Article  Google Scholar 

  19. He, Z., Wong, M.W.: Localization operators associated to square integrable group representations. Panamer. Math. J. 6(1), 93–104 (1996)

    MathSciNet  MATH  Google Scholar 

  20. Hleili, K., Omri, S.: The Littlwood–Paley g-function associated with the spherical mean operator. Mediterr. J. Math. 10(2), 887–907 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  21. John, F.: Plane Waves and Spherical Means Applied to Partial Differential Equations. Interscience, New York (1955)

    Google Scholar 

  22. Lamouchi, H., Majjaouli, B., Omri, S.: Localization of orthonormal sequences in the spherical mean setting. Mediterr. J. Math. 13(4), 1855–1870 (2016)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  24. Ma, B., Wong, M.W.: \(L^{p}\)-boundedness of wavelet multipliers. Hokkaido Math. J. 33, 637–645 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  25. Mejjaoli, H., Othmani, Y.: Uncertainty principles for the generalized Fourier transform associated with spherical mean operator. Anal. Theory Appl. 29, 309–332 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  26. Mejjaoli, H., Othmani, Y.: \(L^{p}\) quantitative uncertainty principles for the generalized Fourier transform associated with the spherical mean operator. Commun. Math. Anal. 18(1), 83–99 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Mejjaoli, H.: Spectral theorems associated with the spherical mean two-wavelet multipliers. Integ. Transf. Spec. Funct. 29(8), 641–662 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  28. Mejjaoli, H., Trimèche, K.: Spectral theorems associated with the spherical mean two-wavelet localization operators. Mediterr. J. Math. 15, Article number: 112 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  29. Mejjaoli, H., Ben Hamadi, N., Omri, S.: Localization operators, time frequency concentration and quantitative-type uncertainty for the continuous wavelet transform associated with spherical mean operator. International Journal of Wavelets. Multiresol. Inf. Process. 17(4), 1950022 (2019)

    Article  MATH  Google Scholar 

  30. Mselhi, N., Rachdi, L.T.: Heisenberg–Pauli–Weyl uncertainty principle for the spherical mean operator. Mediterr. J. Math. 7(2), 169–194 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  31. Nessibi, M.M., Rachdi, L.T., Trimèche, K.: Ranges and inversion formulas for spherical mean operator and its dual. J. Math. Anal. Appl. 196, 861–884 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  32. Omri, S.: Uncertainty principle in terms of entropy for the spherical mean operator. J. Math. Inequal. 5(4), 473–490 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  33. Rachdi, L.T., Trimèche, K.: Weyl transforms associated with the spherical mean operator. Anal. Appl. 1(2), 141–164 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  34. Ramanathan, J., Topiwala, P.: Time–frequency localization via the Weyl correspondence. SIAM J. Math. Anal. 24(5), 1378–1393 (1993)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

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

    Google Scholar 

  37. Zhao, J., Peng, L.: Wavelet and Weyl transforms associated with the spherical mean operator. Integr. Equ. Oper. Theory 50, 279–290 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  38. Wong, M.W.: Localization operators on the Weyl–Heisenberg group. In: Pathak, R.S. (ed.) Geometry, Analysis and Applications, pp. 303–314. Singapore, World-Scientific (2001)

    Google Scholar 

  39. Wong, M.W.: \(L^{p}\) boundedness of localization operators associated to left regular representations. Proc. Am. Math. Soc. 130, 2911–2919 (2002)

    Article  MATH  Google Scholar 

  40. Wong, M.W.: Wavelet Transforms and Localization Operators, p. 136. Springer, Berlin (2002)

    Google Scholar 

Download references

Acknowledgements

The authors are deeply indebted to the referees for providing constructive comments and helps in improving the contents of this article. First author thanks professors K. Trimèche and M.W. Wong for their help.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Slim Omri.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mejjaoli, H., Omri, S. Boundedness and Compactness of the Spherical Mean Two-Wavelet Localization Operators. Bull Braz Math Soc, New Series (2021). https://doi.org/10.1007/s00574-020-00241-6

Download citation

Keywords

  • Spherical mean operator
  • Spherical mean two-wavelt localization operators
  • Schatten–von Neumann class

Mathematics Subject Classification

  • Primary 44A05
  • Secondary 42B10
  • 42B15