Skip to main content
SpringerLink
Log in

\((\varvec{k},\varvec{a})\)-generalized wavelet transform and applications

  • Published:
Journal of Pseudo-Differential Operators and Applications Aims and scope Submit manuscript

Abstract

We introduce the notion of the (k, a)-generalized wavelet transform. Particular cases of such generalized wavelet transform are the classical and the Dunkl wavelet transforms. The restriction of the (k, a)-generalized wavelet transform to radial functions is given by the generalized Hankel wavelet transform. We prove for this new transform Plancherel’s formula, inversion theorem and a Calderón reproducing formula. As applications on the (k, a)-generalized wavelet transform, we give some applications of the theory of reproducing kernels to the Tikhonov regularization on the generalized Sobolev spaces. Next, we study the generalized wavelet localization operators.

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

Fig. 1

Similar content being viewed by others

References

  1. Atakishiyev, N., Wolf, K.B.: Fractional Fourier–Kravchuk transform. J. Opt. Soc. Am. A 14, 1467–1477 (1997)

    Article  MathSciNet  Google Scholar 

  2. Ben Said, S., Kobayashi, T., Ørsted, B.: Laguerre semigroup and \((k, a)\)-generalized operators. Compos. Math. 148(4), 1265–1336 (2012)

    Article  MathSciNet  Google Scholar 

  3. Ben Said, S.: Strichartz estimates for Schrödinger–Laguerre operators. Semigroup Forum 90, 251–269 (2015)

    Article  MathSciNet  Google Scholar 

  4. Brackx, F., De Schepper, N., Sommen, F.: The Clifford–Fourier transform. J. Fourier Anal. Appl. 11, 669–681 (2005)

    Article  MathSciNet  Google Scholar 

  5. Chui, C.K.: An Introduction to Wavelets. Academic Press, Waltham (1992)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

  8. De Bie, H., Xu, Y.: On the Clifford–Fourier transform. Int. Math. Res. Not. IMRN 22, 5123–5163 (2011)

    MathSciNet  MATH  Google Scholar 

  9. De Bie, H., Ørsted, B., Somberg, P., Souček, V.: Dunkl operators and a family of realizations of osp(1|2). Trans. Am. Math. Soc. 364, 3875–3902 (2012)

    Article  MathSciNet  Google Scholar 

  10. De Bie, H.: The kernel of the radially deformed Fourier transform. Integral Transforms Spec. Funct. 24(12), 1000–1008 (2013)

    Article  MathSciNet  Google Scholar 

  11. De Bie, H.: Clifford algebras, Fourier transforms, and quantum mechanics. Math. Methods Appl. Sci. 35(18), 2198–2228 (2012)

    Article  MathSciNet  Google Scholar 

  12. De Bie, H., Ørsted, B., Somberg, P., Souček, V.: \((k, a)\)-generalized operators and a family of realizations of \(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)\). Trans. Am. Math. Soc. 364(7), 3875–3902 (2012)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Folland, G.B.: Harmonic Analysis in Phase Space. Annals of Mathematics Studies, vol. 122. Princeton University Press, Princeton (1989)

    Book  Google Scholar 

  16. Grossmann, A., Morlet, J.: Decomposition of Hardy functions intosquare integrable wavelets of constant shape. SIAM J. Math. Anal. 15(4), 723–736 (1984)

    Article  MathSciNet  Google Scholar 

  17. Gorbachev, D., Ivanov, V., Tikhonov, S.: Sharp Pitt inequality and logarithmic uncertainty principle for \((k, a)\)-generalized transform in \(L^{2}\). Int. Math. Res. Not. 2016(23), 7179–7200 (2016)

    Article  Google Scholar 

  18. Goupilland, P., Grossmann, A., Morlet, J.: Cycle octave and related transforms in seismic signal analysis. Geoexploration 23, 85–102 (1984–1985)

  19. Holschneider, M.: Wavelets: An Analysis Tool. Clarendon Press, Oxford (1995)

    MATH  Google Scholar 

  20. Howe, R.: The oscillator semigroup. In: Wells, R.O. Jr. (ed.) The Mathematical Heritage of Hermann Weyl (Durham, NC, 1987) Proceedings of Symposia in Pure Mathematics, vol. 48, pp. 61–132. American Mathematical Society, Providence (1988)

  21. Howe, R., Tan, E.C.: Nonabelian Harmonic Analysis. Universitext. Springer, New York (1992)

    Google Scholar 

  22. Jafarov, E.I., Stoilova, N.I., Van der Jeugt, J.: The \(su(2)_{\alpha }\) Hahn oscillator and a discrete Hahn–Fourier transform. J. Phys. A Math. Theor. 44, 355205 (2011)

    Article  Google Scholar 

  23. 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 

  24. Kobayashi, T., Mano, G.: The inversion formula an holomorphic extension of the minimal representation of the conformal group, harmonic analysis, group representations, automorphic forms and invariant theory: in honor of Roger Howe. Word Sci. 2007, 159–223 (2007)

    Google Scholar 

  25. Kobayashi, T., Mano, G.: The Schrödinger model for the minimal representation of the indefinite orthogonal group \(O(p,q)\). Mem. Am. Math. Soc. 212(1000), vi + 132pp (2011)

  26. Koornwinder, T.H.: The continuous wavelet transform. Series in approximations and decompositions. In: Koornwinder, T.H. (ed.) Wavelets: An Elementary Treatment of Theory and Applications, vol. 1, pp. 27–48. World Scientific, Singapore (1993)

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  28. Meyer, Y.: Wavelets and Operators. Press Syndicate of University of Cambridge, Cambridge (1995)

    MATH  Google Scholar 

  29. Mejjaoli, H.: Practical inversion formulas for the Dunkl–Gabor transform on \(\mathbb{R}^{d}\). Integral Transform Spec. Funct. 23(12(4)), 875–890 (2012)

    Article  Google Scholar 

  30. 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 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  33. Saitoh, S.: Integral Transforms, Reproducing Kernels and Their Applications. Pitman Research Notes in Mathematics Series 369. Addison Wesley Longman Ltd (1997)

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

    Article  MathSciNet  Google Scholar 

  35. Trimèche, K.: Generalized Wavelets and Hypergroups. Gordon and Breach Science Publishers, Abingdon-on-Thames (1997)

    MATH  Google Scholar 

  36. Wong, M.W.: Wavelet transforms and localization operators, vol. 136. Springer, Berlin (2002)

    Book  Google Scholar 

Download references

Acknowledgements

The author is deeply indebted to the referees for providing constructive comments and helps in improving the contents of this article. The author thanks the professors K. Trimèche, M.W. Wong, S. Ben Said and C. Beddani for their helps.

Author information

Authors and Affiliations

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

    Hatem Mejjaoli

Authors
  1. Hatem Mejjaoli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hatem Mejjaoli.

Additional information

This paper is dedicated to my mother Naziha Bakri.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mejjaoli, H. \((\varvec{k},\varvec{a})\)-generalized wavelet transform and applications. J. Pseudo-Differ. Oper. Appl. 11, 55–92 (2020). https://doi.org/10.1007/s11868-019-00291-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11868-019-00291-5

Keywords

Mathematics Subject Classification

Search

Navigation