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Uncertainty Principle for Free Metaplectic Transformation

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Abstract

This study devotes to Heisenberg’s uncertainty inequalities of complex-valued functions in two free metaplectic transformation (FMT) domains without the assumption of orthogonality. In our latest work (Zhang in J Fourier Anal Appl 27(4):68, 2021), it is crucial that the FMT needs to be orthogonal for a decoupling of the cross terms. Instead of applying the orthogonality assumption, our current work uses the trace inequality for the product of symmetric matrices and positive semidefinite matrices to address the problem of coupling between cross terms. It formulates two types of lower bounds on the uncertainty product of complex-valued functions for two FMTs. The first one relies on the minimum eigenvalues of \(\textbf{A}_j^{\textrm{T}}\textbf{A}_j-\textbf{B}_j^{\textrm{T}}\textbf{A}_j,\textbf{B}_j^{\textrm{T}}\textbf{A}_j,\textbf{B}_j^{\textrm{T}}\textbf{B}_j-\textbf{B}_j^{\textrm{T}}\textbf{A}_j\), while the other one relies on the minimum eigenvalues of \(\textbf{A}_j^{\textrm{T}}\textbf{A}_j+\textbf{B}_j^{\textrm{T}}\textbf{A}_j,\textbf{B}_j^{\textrm{T}}\textbf{B}_j+\textbf{B}_j^{\textrm{T}}\textbf{A}_j\) and the maximum eigenvalues of \(\textbf{B}_j^{\textrm{T}}\textbf{A}_j\), where \(\textbf{A}_j,\textbf{B}_j\), \(j=1,2\) are the blocks found in symplectic matrices. Also, they are all relying on the covariance and absolute covariance. Sufficient conditions that truly give rise to the lower bounds are obtained. The theoretical results are verified by examples and experiments.

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Acknowledgements

The author sincerely thanks the EiC Prof. Hans G. Feichtinger and anonymous referees for their insightful comments, questions, and suggestions that significantly improved the content of the manuscript and presentation of the results.

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  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Zhichao Zhang

  2. Center for Applied Mathematics of Jiangsu Province, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Zhichao Zhang

  3. Jiangsu International Joint Laboratory on System Modeling and Data Analysis, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Zhichao Zhang

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Communicated by Maurice De Gosson.

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This work was supported by the National Natural Science Foundation of China under Grant No 61901223 and the Jiangsu Planned Projects for Postdoctoral Research Funds under Grant No 2021K205B.

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Zhang, Z. Uncertainty Principle for Free Metaplectic Transformation. J Fourier Anal Appl 29, 71 (2023). https://doi.org/10.1007/s00041-023-10052-0

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