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Keywords:
Fourier-Wigner transform; Wigner transform; Moyal identity; time and frequency marginal condition; reconstruction formula; symbol; Weyl transform; kernel; Hilbert-Schmidt operator; Weyl calculus
Fourier-Wigner transform; Wigner transform; Moyal identity; time and frequency marginal condition; reconstruction formula; symbol; Weyl transform; kernel; Hilbert-Schmidt operator; Weyl calculus
Summary:
The periodic Wigner transform is introduced. We show that most of the properties of the Euclidean Wigner transform are satisfied in this new setting. Using the periodic Wigner transform, we define the periodic Weyl transform. $L^2$-boundedness of periodic Weyl transforms are investigated. We give a necessary and sufficient condition on the symbol to ensure that the corresponding periodic Weyl transform is a Hilbert-Schmidt operator. We show that the product of two periodic Weyl transforms and the adjoint of a periodic Weyl transform are again periodic Weyl transforms. The connection between pseudo-differential operators on ${\mathbb S}^1$ and periodic Weyl transforms is given.
References:
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