The circular Bedrosian identity for translation‐invariant operators: existence and characterization |
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Authors: | Wei Hu Rongrong Lin Haizhang Zhang |
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Affiliation: | 1. School of Mathematics and Computational Science, Sun Yat‐sen University, Guangzhou, People's Republic of China;2. School of Mathematics and Computational Science and Guangdong Province Key Laboratory of Computational Science, Sun Yat‐sen University, Guangzhou, People's Republic of China |
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Abstract: | The analytic signal method via the circular Hilbert transform is a critical tool in the time–frequency analysis of signals of finite duration. The circular Bedrosian identity is of major theoretical and practical value in the method. The identity holds whenever the Fourier coefficients of f,g∈L2([?π,π]) are respectively supported on A = [?n,m] and for some non‐negative integers 0≤n,m≤+∞. In this note, we investigate the existence of such an identity for a general‐bounded linear translation‐invariant operator on L2([?π,π]d) and for general support sets . We give an insightful geometric characterization of the support sets for the existence. In addition, we find all the support sets for the partial Hilbert transforms. Copyright © 2015 John Wiley & Sons, Ltd. |
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Keywords: | the Bedrosian identity bounded linear translation‐invariant operators the circular Hilbert transform |
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