Skip to main content
SpringerLink
Log in

Reproducing Kernel Theory Associated with the Generalized Stockwell Transform and Applications

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

The aim of this paper is to study for the Dunkl–Stockwell transform some problems of the reproducing kernels theory. In particular, we investigate the Saitoh approach to give some applications of the Tikhonov regularization for some generalized Sobolev spaces associated with the Dunkl–Stockwell transform and we study also some time-frequency concentration problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

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

    MathSciNet  MATH  Google Scholar 

  2. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)

    MathSciNet  MATH  Google Scholar 

  3. Ben Hamadi, N., Hafirassou, Z., Herch, H.: Uncertainty principles for the Hankel–Stockwell transform. J. Pseudo-Differ. Oper. Appl. 11(2), 543–564 (2020)

    MathSciNet  MATH  Google Scholar 

  4. Ben Hamadi, N., Hafirassou, Z.: Local Price uncertainty principle and time-frequency localization operators for the Hankel–Stockwell transform. Int. J. Wavelets Multiresolut. Inf. Process. 18, 2050050 (2020)

    MathSciNet  MATH  Google Scholar 

  5. Ben Hamadi, N., Hafirassou, Z.: Amrein–Berthier and Logvinenko–Sereda uncertainty principles for the Hankel–Stockwell transform. J. Pseudo-Differ. Oper. Appl. bf12(4), 1–23 (2021)

    MathSciNet  MATH  Google Scholar 

  6. Bouzeffour, F.: On the norm of the \(L^{p}\)Dunkl transform. Appl. Anal. 94(4), 761–779 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Cai, H.-P., He, Z.H., Huang, D.J.: Seismic data denoising based on mixed time-Frequency methods. Appl. Geophys. 8, 319–327 (2011)

    Google Scholar 

  8. Castro, L.P., Saitoh, S., Sawano, Y., Simões, A.M.: General inhomogeneous discrete linear partial differential equations with constant coefficients on the whole spaces. Complex Anal. Oper. Theory 6, 307–324 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Catana, V.: Schatten-von Neumann norm inequalities for two-wavelet localization operators associated with \(\beta \)-Stockwell transforms. Appl Anal. 91, 503–515 (2012)

    MathSciNet  MATH  Google Scholar 

  10. de Jeu, M.F.E.: The Dunkl transform. Invent. Math. 113, 147–162 (1993)

    MathSciNet  MATH  Google Scholar 

  11. Djurovic, I., Sejdic, E., Jiang, J.: Frequency-based window width optimization for \(S-\)transform. AEU - Int. J. Electron. Commun. 62, 245–250 (2008)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Du, J., Wong, M.W., Zhu, H.: Continuous and discrete inversion formulas for the Stockwell transform. Integr. Transforms Spec. Funct. 18, 537–543 (2007)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  15. Dunkl, C.F.: Hankel transforms associated to finite reflection groups. Contemp. Math. 138, 123–138 (1992)

    MathSciNet  MATH  Google Scholar 

  16. Gabor, D.: Theory of communication. Part 1: The analysis of information. J. Inst. Electr. Eng.-Part III Radio Commun. Eng. 93(26), 429–441 (1946)

  17. Ghobber, S., Jaming, P.: Uncertainty principles for integral orperators. Studia Math. 220, 197–220 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Ghobber, S.: Dunkl-Gabor transform and time-frequency concentration. Czechoslov. Math. J. 65(1), 255–270 (2015)

    MathSciNet  MATH  Google Scholar 

  19. Ghobber, S.: Time-frequency concentration and localization operators in the Dunkl setting. J. Pseudo-Differ. Oper. Appl. 7(3), 431–449 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Ghobber, S.: Some results on wavelet scalograms. Int. J. Wavelets Multiresolut. Inf. Process. 15(3), 1750019 (2017)

    MathSciNet  MATH  Google Scholar 

  21. Havin, V., Jöricke, B.: The Uncertainty Principle in Harmonic Analysis, vol. 24. Springer, Berlin (1994)

    MATH  Google Scholar 

  22. Kawazoe, T., Mejjaoli, H.: Uncertainty principles for the Dunkl transform. Hiroshima Math. J. 40(2), 241–268 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Malinnikova, E.: Orthonormal sequences in \(L^{2} ({{\mathbb{R} }}^{d})\) and time frequency localization. J. Fourier Anal. Appl. 16(6), 983–1006 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Mejjaoli, H., Trimèche, K.: Hypoellipticity and hypoanalyticity of the Dunkl Laplacian operator. Integ. Transf. Special Funct. 15(6), 523–548 (2004)

    MathSciNet  MATH  Google Scholar 

  25. Mejjaoli, H., Trimèche, K.: Spectrum of functions for the Dunkl transform on \({{\mathbb{R} }}^{d}\). Fract. Cal. Appl. Anal. 10(1), 19–38 (2007)

    MATH  Google Scholar 

  26. Mejjaoli, H., Sraieb, N.: Uncertainty principles for the continuous Dunkl wavelet transform and the Dunkl continuous Gabor transform. Mediterr. J. Math. 5, 443–466 (2008)

    MathSciNet  MATH  Google Scholar 

  27. Mejjaoli, H.: Practical inversion formulas for the Dunkl–Gabor transform on \({{\mathbb{R} }}^{d}\). Integr. Transf. Spec. Funct. 23(12), 875–890 (2012)

    MATH  Google Scholar 

  28. Mejjaoli, H., Trimèche, K.: Time-Frequency Concentration, Heisenberg Type Uncertainty Principles and Localization Operators for the Continuous Dunkl Wavelet Transform on \({\mathbb{R} }^{d}\). Mediterran. J. Math. 14(146), 1 (2017)

    MATH  Google Scholar 

  29. Mejjaoli, H.: Dunkl–Stockwell transform and its applications to the time-frequency analysis. J. Pseudo-Differ. Oper. Appl. 12(2), 1–59 (2021)

    MathSciNet  MATH  Google Scholar 

  30. Riba, L., Wong, M.W.: Continuous inversion formulas for multi-dimensional modified Stockwell transforms. Integral Transform. Spec. Funct. 26(1), 9–19 (2015)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  32. Rösler, M., Voit, M.: Markov processes related with Dunkl operators. Adv. Appl. Math. 21(4), 575–643 (1998)

    MathSciNet  MATH  Google Scholar 

  33. Rösler, M.: Positivity of Dunkl’s intertwining operator. Duke Math. J. 98, 445–463 (1999)

    MathSciNet  MATH  Google Scholar 

  34. Rösler, M.: A positive radial product formula for the Dunkl kernel. Trans. Am. Math. Soc. 355, 2413–2438 (2003)

    MathSciNet  MATH  Google Scholar 

  35. Saitoh, S.: Theory of Reproducing Kernels and its Applications. Longman Scientific Technical, Harlow (1988)

    MATH  Google Scholar 

  36. Saitoh, S.: Inequalities in the Most Simple Sobolev Space and Convolutions of \(L^{2}\) Functions with Weights. Proc. Am. Math. Soc. 118(2), 515–520 (1993)

    MATH  Google Scholar 

  37. Saitoh, S.: One approach to some general integral transforms and its applications. Integral Transform. Spec. Funct. 3, 49–84 (1995)

    MathSciNet  MATH  Google Scholar 

  38. Saitoh, S.: Integral Transforms, Reproducing Kernels and their Applications. Pitman Research Notes in Math. Series 369, Addison Wesley Longman, Harlow (1997)

  39. Saitoh, S.: Best approximation, Tikhonov regularization and reproducing kernels. Kodai Math. J. 28(2), 359–367 (2005)

    MathSciNet  MATH  Google Scholar 

  40. Schwartz, L.: Sous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux associés (noyaux reproduisants). J. Anal. Math. 13, 115–256 (1964)

    MATH  Google Scholar 

  41. Shah, F.A., Tantary, A.Y.: Non-isotropic angular Stockwell transform and the associated uncertainty principles. Appl Anal. 100(4), 835–859 (2021)

    MathSciNet  MATH  Google Scholar 

  42. Shah, F.A., Tantary, A.Y.: Linear canonical Stockwell transform. J. Math. Anal. Appl. 484(1), 123673 (2020). https://doi.org/10.1016/j.jmaa.2019.123673

    Article  MathSciNet  MATH  Google Scholar 

  43. Srivastava, H.M., Shah, F.A., Tantary, A.Y.: A family of convolution-based generalized Stockwell transforms. J. Pseudo-Differ. Oper. Appl. 11, 1505–1536 (2020)

    MathSciNet  MATH  Google Scholar 

  44. Stockwell, R.G., Mansinha, L., Lowe, R.P.: Localization of the complex spectrum: the \(S\) transform. IEEE Trans. Signal Process. 44, 998–1001 (1996)

    Google Scholar 

  45. Thangavelu, S., Xu, Y.: Convolution operator and maximal functions for Dunkl transform. J. d’Analyse Mathematique 97, 25–56 (2005)

    MathSciNet  MATH  Google Scholar 

  46. Trimèche, K.: Paley-Wiener theorems for Dunkl transform and Dunkl translation operators. Integ. Transf. Spec. Funct. 13, 17–38 (2002)

    MathSciNet  MATH  Google Scholar 

  47. Trimèche, K.: Inversion of the Dunkl intertwining operator and its dual using Dunkl wavelet. Rocky Mount. J. Math. 32(2), 889–918 (2002)

    MathSciNet  MATH  Google Scholar 

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

  49. Wong, M.W.: Wavelet Transforms and Localization Operators, vol. 136. Springer (2002)

Download references

Acknowledgements

The authors are deeply indebted to the referees for providing constructive comments and helps in improving the contents of this article. Hatem Mejjaoli dedicate this paper to the Emeritus Professors Saburou Saitoh and Khalifa Trimèche.

Funding

This work was supported through the Annual Funding track by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Project No. GRANT602].

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, College of Science, King Faisal University, PO. Box: 400, Al-Ahsa, 31982, Saudi Arabia

    Saifallah Ghobber

  2. Department of Mathematics, College of Sciences, Taibah University, PO. Box: 30002, Al-Madinah AL-Munawarah, Saudi Arabia

    Hatem Mejjaoli

Authors
  1. Saifallah Ghobber
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hatem Mejjaoli
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Conceptualization: S.G. and H.M.; Methodology: S.G. and H.M.; Validation: S.G. and H.M.; Formal analysis: S.G. and H.M.; Investigation: S.G. and H.M.; Resources: S.G. and H.M.; Writing original draft preparation: H.M.; Writing-review and editing: S.G.; Project administration: S.G.; funding acquisition: S.G.

Corresponding author

Correspondence to Hatem Mejjaoli.

Ethics declarations

Conflict of interest

The authors have no conflicts of interest to declare.

Ethical Approval

Not applicable.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghobber, S., Mejjaoli, H. Reproducing Kernel Theory Associated with the Generalized Stockwell Transform and Applications. Complex Anal. Oper. Theory 17, 106 (2023). https://doi.org/10.1007/s11785-023-01407-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11785-023-01407-y

Keywords

Mathematics Subject Classification

Search

Navigation