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

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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