Czechoslovak Mathematical Journal, first online, pp. 1-16
Periodic analogs of Wigner transforms and Weyl transforms
Shahla Molahajloo, Man Wah Wong
Received September 1, 2024. Published online March 3, 2025.
Abstract: 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.
Keywords: Fourier-Wigner transform; Wigner transform; Moyal identity; time and frequency marginal condition; reconstruction formula; symbol; Weyl transform; kernel; Hilbert-Schmidt operator; Weyl calculus
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Affiliations: Shahla Molahajloo, Faculty of Applied Science and Technology, Sheridan College, Davis Campus, 7899 McLaughlin Road, Brampton, Ontario L6Y 5H9, Canada, e-mail: smollaha@gmail.com; Man Wah Wong (corresponding author), Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada, e-mail: mwwong@yorku.ca
