Wigner-Weyl′s transform and its contraction

Shahla Molahajloo, Man Wah Wong

Dominik Bayer, Elena Cordero, Karlheinz Gröchenig +1 more

Man Wah Wong

The Fourier–Wigner transform of two functions is first defined and is given an alternative expression. The Wigner transform is then defined to be the Fourier transform of the Fourier–Wigner transform. The Moyal identity, which is an analog of the Plancherel formula, is then proved for the Wigner transform. The Weyl transform with a given symbol is then defined in terms of the Wigner transform. Another formula for the Weyl transform with a given symbol is then proved and used to establish the fact that the product of two Weyl transforms is also a Weyl transform with symbol given by a twisted convolution of the two given symbols.

M. Kira, S. W. Koch

Amit Kumar, Akhilesh Prasad, Pankaj Jain

The main objective of the paper is to generalize and enrich the Weyl transform by introducing the Weyl correspondence in the linear canonical transform (LCT) domain. In this paper, we propose the linear canonical-Wigner transform in harmonic analysis of phase space along with the admissible Wigner-Ville distribution (WVD) and Weyl transform in the LCT domain and discuss some useful results. Further we establish the relationship between the Wigner-Ville distribution and the Weyl transform in the LCT domain.

M.W. Wong

Wigner transforms and Weyl transforms are not only useful as tools in this book, but also arise in many different contexts in mathematics. For this reason, we find it worthwhile to present them in the setting of Rn. The books [4, 40] contain much more information on these topics.