Abstract
We consider in this paper Wigner representations Wig τ 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 Wig Φ. We give basic properties of Wig Φ as subclasses of the general Cohen class.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
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)
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)
Cohen L.: Time–frequency distributions—a review. Proc. IEEE 77(7), 941–981 (1989)
Cohen L.: Time–frequency Analysis. Prentice Hall Signal Proceeding Series, New Jersey (1995)
Gatteschi, L.: Funzioni Speciali, Unione Tipografica Editrice Torinese (1973)
Gröchenig K.: Foundations of Time–Frequency Analysis. Birkhäuser, Boston (2001)
Janssen A.J.A.: Proof of a conjecture on the supports of Wigner distributions. J. Fourier Anal. Appl. 4(6), 723–726 (1998)
Lieb E.H.: Integral bounds for radar ambiguity functions and Wigner distributions. J. Math. Phys. 31(3), 594–599 (1990)
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)
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)
Shubin M.A.: Pseudodifferential Operators and Spectral Theory, 2nd edn. Springer, Berlin (2001)
Wong M.W.: Weyl Transforms. Springer, Berlin (1998)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
The project is supported by WWS Projects of University of Turin, Italy.
Rights and permissions
About this article
Cite this article
Boggiatto, P., Cuong, B.K., De Donno, G. et al. Weighted integrals of Wigner representations. J. Pseudo-Differ. Oper. Appl. 1, 401–415 (2010). https://doi.org/10.1007/s11868-010-0018-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11868-010-0018-x