Advertisement

Acta Mathematica Vietnamica

, Volume 42, Issue 1, pp 129–147 | Cite as

Analytical and Numerical Approximation Formulas on the Dunkl-Type Fock Spaces

  • Fethi SoltaniEmail author
  • Akram Nemri
Article
  • 120 Downloads

Abstract

In this work, we establish some versions of Heisenberg-type uncertainty principles for the Dunkl-type Fock space \(F_{k}(\mathbb {C}^{d})\). Next, we give an application of the classical theory of reproducing kernels to the Tikhonov regularization problem for operator \(L:F_{k}(\mathbb {C}^{d})\rightarrow H\), where H is a Hilbert space. Finally, we come up with some results regarding the Tikhonov regularization problem and the Heisenberg-type uncertainty principle for the Dunkl-type Segal-Bargmann transform \(\mathcal {B}_{k}\). Some numerical applications are given.

Keywords

Dunkl-type Fock spaces Bounded operators Extremal functions Tikhonov regularization Uncertainty principles 

Mathematics Subject Classification (2010)

32A15 32A36 46E22 

Notes

Acknowledgments

The authors thank the Deanship of Scientific Research-Jazan University for its support in financing this research project SABIC 2-588-36.

References

  1. 1.
    Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform, Part I. Comm. Pure Appl. Math. 14, 187–214 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ben Said, S., Ørsted, B.: Segal-Bargmann transforms associated with Coxeter groups. Math. Ann. 334, 281–323 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berger, C.A., Coburn, L.A.: Toeplitz operators on the Segal-Bargmann space. Trans. Am. Math. Soc. 301, 813–829 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cho, H., Zhu, K.: Fock-Sobolev spaces and their Carleson measures. J. Funct. Anal. 263, 2483–2506 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cholewinski, F.M.: Generalized Fock spaces and associated operators. SIAM J. Math. Anal. 15, 177–202 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167–183 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dunkl, C.F.: Integral kernels with reflection group invariance. Can. J. Math. 43, 1213–1227 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dunkl, C.F.: Hankel transforms associated to finite reflection groups. Contemp. Math. 138, 123–138 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Folland, G.: Harmonic analysis on phase space. In: Annals of Mathematics Studies, vol. 122. Princeton University Press, Princeton (1989)Google Scholar
  11. 11.
    Gröchenig, K.: Foundations of time-frequency analysis. Birkhäuser, Boston (2001)CrossRefzbMATHGoogle Scholar
  12. 12.
    de Jeu, M.F.E.: The Dunkl transform. Inv. Math. 113, 147–162 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kimeldorf, G.S., Wahba, G.: Some results on Tchebycheffian spline functions. J. Math. Anal. Appl. 33, 82–95 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lapointe, L., Vinet, L.: Exact operator solution of the Calogero- Sutherland model. Comm. Math. Phys. 178, 425–452 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Matsuura, T., Saitoh, S., Trong, D.D.: Inversion formulas in heat conduction multidimensional spaces. J. Inv. Ill-posed Problems 13, 479–493 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Matsuura, T., Saitoh, S.: Analytical and numerical inversion formulas in the Gaussian convolution by using the Paley-Wiener spaces. Appl. Anal. 85, 901–915 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rösler, M.: Generalized Hermite polynomials and the heat equation for Dunkl operators. Comm. Math. Phys. 192, 519–542 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rösler, M.: An uncertainty principle for the Dunkl transform. Bull. Austral. Math. Soc. 59, 353–360 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rösler, M.: Positivity of Dunkl’s intertwining operator. Duke Math. J. 98, 445–463 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Opdam, E.M.: Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group. Compos. Math. 85(3), 333–373 (1993)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Saitoh, S.: Hilbert spaces induced by Hilbert space valued functions. Proc. Am. Math. Soc. 89, 74–78 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Saitoh, S.: The Weierstrass transform and an isometry in the heat equation. Appl. Anal. 16, 1–6 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Saitoh, S.: Approximate real inversion formulas of the Gaussian convolution. Appl. Anal. 83, 727–733 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Saitoh, S.: Best approximation, Tikhonov regularization and reproducing kernels. Kodai Math. J. 28, 359–367 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sifi, M., Soltani, F.: Generalized Fock spaces and Weyl relations for the Dunkl kernel on the real line. J. Math. Anal. Appl. 270, 92–106 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Soltani, F.: Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel. Pacific J. Math. 214, 379–397 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Soltani, F.: Inversion formulas in the Dunkl-type heat conduction on \(\mathbb {R}^{d}\). Appl. Anal. 84, 541–553 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Soltani, F.: Best approximation formulas for the Dunkl L 2-multiplier operators on \(\mathbb {R}^{d}\). Rocky Mountain J. Math. 42, 305–328 (2012)CrossRefzbMATHGoogle Scholar
  29. 29.
    Soltani, F.: Multiplier operators and extremal functions related to the dual Dunkl-Sonine operator. Acta Math. Sci. 33B(2), 430–442 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Soltani, F.: Heisenberg-Pauli-Weyl uncertainty inequality for the Dunkl transform on \(\mathbb {R}^{d}\). Bull. Austral. Math. Soc. 87, 316–325 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Soltani, F.: A general form of Heisenberg-Pauli-Weyl uncertainty inequality for the Dunkl transform. Int. Trans. Spec. Funct. 24(5), 401–409 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Soltani, F.: An L p Heisenberg-Pauli-Weyl uncertainty principle for the Dunkl transform. Konuralp J. Math. 2(1), 1–6 (2014)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Soltani, F.: Dunkl multiplier operators and applications. Int. Trans. Spec. Funct. 25(11), 898–908 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Yamada, M., Matsuura, T., Saitoh, S.: Representations of inverse functions by the integral transform with the sign kernel. Frac. Calc. Appl. Anal. 2, 161–168 (2007)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Zhu, K.: Analysis on Fock Spaces. Springer-Verlag, New York (2012)CrossRefzbMATHGoogle Scholar
  36. 36.
    Zhu, K.: Uncertainty principles for the Fock space. Preprint 2015, arXiv:1501.02754V1

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceJazan UniversityJazanSaudi Arabia

Personalised recommendations