Spectral theorems associated with the (ka)-generalized wavelet multipliers

  • Hatem Mejjaoli


We introduce the notion of the (ka)-generalized wavelet multipliers. Particular cases of such generalized wavelet multipliers are the classical and Dunkl wavelet multipliers. The restriction of the (ka)-generalized wavelet multipliers to radial functions is given by the generalized Hankel wavelet multiplier. We study the boundedness, Schatten class properties of the (ka)-generalized wavelet multipliers and we give them trace formula. We prove that the generalized Landau–Pollak–Slepian operator is a (ka)-generalized wavelet multiplier. Next, we give results on the boundedness and compactness of (ka)-generalized wavelet multipliers on \(L^{p}_{k,a}(\mathbb {R}^{d})\), \(1 \le p \le \infty \).


\((k, a)\)-Laguerre semigroup \((k, a)\)-generalized Fourier transform \((k, a)\)-generalized multipliers \((k, a)\)-generalized two-wavelet multipliers Schatten-von Neumann class 

Mathematics Subject Classification

Primary 47G10 Secondary 42B10 47G30 



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 and S. Ben Said for their helps.


  1. 1.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Cambridge (1988)zbMATHGoogle Scholar
  2. 2.
    Ben Said, S., Kobayashi, T., Ørsted, B.: Laguerre semigroup and Dunkl operators. Compos. Math. 148(04), 1265–1336 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ben Said, S.: Strichartz estimates for Schrödinger–Laguerre operators. Semigroup Forum 90, 251–269 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Calderon, J.P.: Intermediate spaces and interpolation, the complex method. Studia Mathematica 24, 113–190 (1964)MathSciNetCrossRefGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefGoogle Scholar
  6. 6.
    De Bie, H.: The kernel of the radially deformed Fourier transform. Integr. Transform. Spec. Funct. 24(12), 1000–1008 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    De Bie, H.: Clifford algebras, Fourier transforms, and quantum mechanics. Math. Methods Appl. Sci. 35(18), 2198–2228 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    De Bie, H., Ørsted, B., Somberg, P., Souček, V.: Dunkl operators and a family of realizations of \(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)\). Trans. Am. Math. Soc. 364(7), 3875–3902 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Du, J., Wong, M.W.: Traces of wavelet multipliers. C. R. Math. Rep. Acad. Sci. Can. 23, 148–152 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167–183 (1989)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dunkl, C.F.: Hankel transforms associated to finite reflection groups. In: Proceedings of the Special Session on Hypergeometric Functions on Domains of Positivity, Jack Polynomials and Applications (Tampa, FL, 1991), Contemp. Math. 138, 123–138 (1992)Google Scholar
  12. 12.
    Folland, G.B.: Introduction to Partial Differential Equations, 2nd edn. Princeton University Press, Princeton (1995)zbMATHGoogle Scholar
  13. 13.
    Gorbachev, D., Ivanov, V., Tikhonov, S.: Sharp Pitt inequality and logarithmic uncertainty principle for Dunkl transform in \(L^{2}\). Int. Math. Res. Not. 2016(23), 7179–7200 (2016)CrossRefGoogle Scholar
  14. 14.
    He, Z., Wong, M.W.: Wavelet multipliers and signals. J. Austral. Math. Soc. Ser. B 40, 437–446 (1999)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Howe, R.: The oscillator semigroup. The mathematical heritage of Hermann Weyl (Durham, NC, 1987), 61–132, Proc. Sympos. Pure Math. 48, Amer. Math. Soc., Providence, RI (1988)Google Scholar
  16. 16.
    Johansen, T.-R.: Weighted inequalities and uncertainty principles for the \((k, a)\)-generalized Fourier transform. Int. J. Math. 27(3), 1650019 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    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
  18. 18.
    Kobayashi, T., Mano, G.: The Schrödinger model for the minimal representation of the indefinite orthogonal group \(O(p,q)\). vi + pp. 132, Mem. Amer. Math. Soc. vol. 212(1000), (2011)Google Scholar
  19. 19.
    Liu, L.: A trace class operator inequality. J. Math. Anal. Appl. 328, 1484–1486 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Rösler, M.: Positivity of Dunkl’s intertwining operator. Duke Math. J. 98, 445–463 (1999)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Stein, E.M.: Interpolation of linear operators. Trans. Am. Math. Soc. 83, 482–492 (1956)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Wong, M.W.: Wavelet Transforms and Localization Operators, vol. 136. Springer, Berlin (2002)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesTaibah UniversityAl Madinah Al MunawarahSaudi Arabia

Personalised recommendations