Article
| Title: | Periodic analogs of Wigner transforms and Weyl transforms (English) |
| Author: | Molahajloo, Shahla |
| Author: | Wong, Man Wah |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 2 |
| Year: | 2025 |
| Pages: | 629-644 |
| Summary lang: | English |
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| Category: | math |
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| 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. (English) |
| Keyword: | Fourier-Wigner transform |
| Keyword: | Wigner transform |
| Keyword: | Moyal identity |
| Keyword: | time and frequency marginal condition |
| Keyword: | reconstruction formula |
| Keyword: | symbol |
| Keyword: | Weyl transform |
| Keyword: | kernel |
| Keyword: | Hilbert-Schmidt operator |
| Keyword: | Weyl calculus |
| MSC: | 47F05 |
| MSC: | 47G30 |
| DOI: | 10.21136/CMJ.2025.0369-24 |
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| Date available: | 2025-05-20T11:49:15Z |
| Last updated: | 2025-05-26 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152962 |
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| Reference: | [1] Cohen, L.: The Weyl Operator and its Generalization.Pseudo-Differential Operators. Theory and Applications 9. Birkhäuser, Basel (2013). Zbl 1393.47001, MR 3012699, 10.1007/978-3-0348-0294-9 |
| Reference: | [2] Dasgupta, A., Wong, M. W.: Pseudo-differential operators on the affine group.Pseudo- Differential Operators: Groups, Geometry and Applications Trends in Mathematics. Birkhäuser, Basel (2017), 1-14. Zbl 1377.47018, MR 3708360, 10.1007/978-3-319-47512-7_1 |
| Reference: | [3] Duan, X., Wong, M. W.: Pseudo-differential operators for Weyl transforms.Sci. Bull., Ser. A, Appl. Math. Phys., Politeh. Univ. Buchar. 75 (2013), 3-12. Zbl 1299.47095, MR 3161943 |
| Reference: | [4] Molahajloo, S.: Pseudo-differential operators on $\Bbb{Z}$.Pseudo-Differential Operators. Complex Analysis and Partial Differential Equations Operator Theory: Advances and Applications 205. Birkhäuser, Basel (2010), 213-221. Zbl 1227.47028, MR 2664583, 10.1007/978-3-0346-0198-6_12 |
| Reference: | [5] Molahajloo, S.: Pseudo-differential operators, Wigner transforms and Weyl transforms on the Poincaré unit disk.Complex Anal. Oper. Theory 12 (2018), 811-833. Zbl 1466.47034, MR 3770371, 10.1007/s11785-018-0777-6 |
| Reference: | [6] Molahajloo, S., Wong, M. W.: Pseudo-differential operators on $\Bbb{S}^1$.New Developments in Pseudo-Differential Operators Operator Theory: Advances and Applications 189. Birkhäuser, Basel (2009), 297-306. Zbl 1210.47073, MR 2509104, 10.1007/978-3-7643-8969-7_15 |
| Reference: | [7] Molahajloo, S., Wong, M. W.: Ellipticity, Fredholmness and spectral invariance of pseudo- differential operators on $\Bbb{S}^1$.J. Pseudo-Differ. Oper. Appl. 1 (2010), 183-205. Zbl 1222.47075, MR 2679899, 10.1007/s11868-010-0010-5 |
| Reference: | [8] Molahajloo, S., Wong, M. W.: Discrete analogs of Wigner transforms and Weyl transforms.Analysis of Pseudo-Differential Operators Trends in Mathematics. Birkhäuser, Cham (2019), 1-20. Zbl 07121761, MR 4012640, 10.1007/978-3-030-05168-6_1 |
| Reference: | [9] Peng, L., Zhao, J.: Weyl transforms associated with the Heisenberg group.Bull. Sci. Math. 132 (2008), 78-86. Zbl 1136.43008, MR 2417612, 10.1016/j.bulsci.2007.07.002 |
| Reference: | [10] Peng, L., Zhao, J.: Weyl transforms on the upper half plane.Rev. Mat. Complut. 23 (2010), 77-95. Zbl 1202.44003, MR 2578572, 10.1007/s13163-009-0013-z |
| Reference: | [11] Tate, T.: Weyl pseudo-differential operator and Wigner transform on the Poincaré disk.Ann. Global Anal. Geom. 22 (2002), 29-48. Zbl 1022.58011, MR 1916978, 10.1023/A:1016253829938 |
| Reference: | [12] Wong, M. W.: Weyl Transforms.Universitext. Springer, New York (1998). Zbl 0908.44002, MR 1639461, 10.1007/b98973 |
| Reference: | [13] Wong, M. W.: Discrete Fourier Analysis.Pseudo-Differential Operators. Theory and Applications 5. Birkhäuser, Basel (2011). Zbl 1226.42001, MR 2809393, 10.1007/978-3-0348-0116-4 |
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