Weighted integrals of Wigner representations

  • Paolo Boggiatto
  • Bui Kien Cuong
  • Giuseppe De Donno
  • Alessandro Oliaro
Article

Abstract

We consider in this paper Wigner representations Wigτ depending on a parameter \({\tau\in[0,1]}\) as defined in Boggiatto et al. (Trans Am Math Soc 362:4955–4981, 2010). Integrating these forms with respect to the parameter τ against a weight function Φ we obtain a new class of time–frequency representations WigΦ. We give basic properties of WigΦ as subclasses of the general Cohen class.

Keywords

Time–frequency representation τ-Wigner distribution Cohen class 

Mathematics Subject Classification (2000)

42B10 47A07 47B38 47G30 

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References

  1. 1.
    Boggiatto P., De Donno G., Oliaro A.: Time–frequency representations of Wigner type and pseudo-differential operators. Trans. Am. Math. Soc. 362(9), 4955–4981 (2010)MATHMathSciNetGoogle Scholar
  2. 2.
    Boggiatto P., De Donno G., Oliaro A.: A class of quadratic time–frequency representations based on the short-time Fourier transform. Oper. Theory Adv. Appl. 172, 235–249 (2006)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Boggiatto P., De Donno G., Oliaro A.: Uncertainty principle, positivity and L p-boundedness for generalized spectrograms. J. Math. Anal. Appl. 355(1), 93–112 (2007)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Cohen L.: Time–frequency distributions—a review. Proc. IEEE 77(7), 941–981 (1989)CrossRefGoogle Scholar
  5. 5.
    Cohen L.: Time–frequency Analysis. Prentice Hall Signal Proceeding Series, New Jersey (1995)Google Scholar
  6. 6.
    Gatteschi, L.: Funzioni Speciali, Unione Tipografica Editrice Torinese (1973)Google Scholar
  7. 7.
    Gröchenig K.: Foundations of Time–Frequency Analysis. Birkhäuser, Boston (2001)MATHGoogle Scholar
  8. 8.
    Janssen A.J.A.: Proof of a conjecture on the supports of Wigner distributions. J. Fourier Anal. Appl. 4(6), 723–726 (1998)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Lieb E.H.: Integral bounds for radar ambiguity functions and Wigner distributions. J. Math. Phys. 31(3), 594–599 (1990)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Mohammed, A., Wong, M.W.: Rihaczek transform and pseudo-differential operators. In: Pseudo-Differential Operators: Partial Differential Equations and Time–Frequency Analysis. Fields Institute Communication, vol. 52, pp. 375–382. American Mathematical Society, Providence (2007)Google Scholar
  11. 11.
    Toft, J.: Hudson’s theorem and rank one operators in Weyl calculus. In: Pseudo-Differential Operators and Related Topics. Operator Theory: Advances and Applications, vol. 164, pp. 153–159. Birkhäuser, Basel (2006)Google Scholar
  12. 12.
    Shubin M.A.: Pseudodifferential Operators and Spectral Theory, 2nd edn. Springer, Berlin (2001)MATHGoogle Scholar
  13. 13.
    Wong M.W.: Weyl Transforms. Springer, Berlin (1998)MATHGoogle Scholar
  14. 14.
    Wong, M.W.: Symmetry-breaking for Wigner transform and L p-boundedness of Weyl transform, In: Ashino, R., Boggiatto, P., Wong, M.W. (eds.). Operational Therotical Advances and Applications: Advances in Pseudo-differential Operators, vol. 155, pp. 107–116. Birkhäuser, Basel (2004)Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Paolo Boggiatto
    • 1
  • Bui Kien Cuong
    • 2
  • Giuseppe De Donno
    • 1
  • Alessandro Oliaro
    • 1
  1. 1.Department of MathematicsUniversity of TurinTurinItaly
  2. 2.Higher Education DepartmentHanoi Pedagogical University 2HanoiVietnam

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