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Uncertainty Principle of Complex-Valued Functions in Specific Free Metaplectic Transformation Domains

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Abstract

This study devotes to uncertainty principles under some specific free metaplectic transformations of complex-valued functions. Two versions of uncertainty inequalities in two orthogonal free metaplectic transformation domains are established. The first one provides a lower bound which is closely related to all the elements in the blocks \(\mathbf {A}_j,\mathbf {B}_j\), \(j=1,2\) found in symplectic matrices, while the other one depends on singular values of these four blocks by focusing on the orthonormal case. The latter is tighter than the existing forms for two categories, including the orthonormal free metaplectic transformation of all functions and two orthonormal free metaplectic transformations of real-valued functions. Conditions that truly reach these lower bounds are deduced. The proposed uncertainty principles can be reduced to the separable case, giving rise to uncertainty inequalities in two separable free metaplectic transformation domains. Examples and simulations are carried out to verify the correctness of the derived results, and finally possible applications in time-frequency analysis and optical system analysis are also given. As a result, this paper partly solved a concern on an extension to complex-valued functions mentioned in the conclusion of our previous work (Zhang in J Fourier Anal Appl 25:2899–2922, 2019).

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Acknowledgements

The author would like to thank the anonymous reviewers for making many useful suggestions to the manuscript.

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Authors and Affiliations

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

    Zhichao Zhang

  2. Faculty of Information Technology, Macau University of Science and Technology, Macau, 999078, China

    Zhichao Zhang

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  1. Zhichao Zhang
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Correspondence to Zhichao Zhang.

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Communicated by Veluma Thangavelu.

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This work was supported by the National Natural Science Foundation of China under Grant No. 61901223, the Natural Science Foundation of Jiangsu Province under Grant No. BK20190769, the Jiangsu Planned Projects for Postdoctoral Research Funds, the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant No. 19KJB510041, the Jiangsu Province High-Level Innovative and Entrepreneurial Talent Introduction Program under Grant No. R2020SCB55, the Macau Young Scholars Program under Grant No. AM2020015, and the Startup Foundation for Introducing Talent of NUIST under Grant No. 2019r024.

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Zhang, Z. Uncertainty Principle of Complex-Valued Functions in Specific Free Metaplectic Transformation Domains. J Fourier Anal Appl 27, 68 (2021). https://doi.org/10.1007/s00041-021-09867-6

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