The free metaplectic transformation (FMT) has gained much popularity in recent times because of its various application in signal processing, paraxial optical systems, digital algorithms, optical encryption and so on. However, the FMT is inadequate for localized analysis of non-transient signals, as such, it is imperative to introduce a unique localized transform coined as the short time free metaplectic transform (STFMT). In this paper, we investigate the STFMT. Firstly, we propose the definition of the STFMT, and provide the time frequency analysis of the proposed transform in the FMT domain. Secondly, we investigate the basic properties of the proposed transform including the reconstruction formula, Moyals formula. The emergence of the STFMT definition and its properties broadens the development of time-frequency representation of higher-dimensional signals theory to a certain extent. Finally, we extend some different uncertainty principles (UP) from quantum mechanics including Liebs, Pitts UP, Heisenbergs uncertainty principle, Hausdorff-Young, Hardys UP, Beurlings UP, Logarithmic UP, and Nazarovs UP which have already been well studied in the FMT domainThe free metaplectic transformation (FMT) has gained much popularity in recent times because of its various application in signal processing, paraxial optical systems, digital algorithms, optical encryption and so on. However, the FMT is inadequate for localized analysis of non-transient signals, as such, it is imperative to introduce a unique localized transform coined as the short time free metaplectic transform (STFMT). In this paper, we investigate the STFMT. Firstly, we propose the definition of the STFMT, and provide the time frequency analysis of the proposed transform in the FMT domain. Secondly, we investigate the basic properties of the proposed transform including the reconstruction formula, Moyals formula. The emergence of the STFMT definition and its properties broadens the development of time-frequency representation of higher-dimensional signals theory to a certain extent. Finally, we extend some different uncertainty principles (UP) from quantum mechanics including Liebs, Pitts UP, Heisenbergs uncertainty principle, Hausdorff-Young, Hardys UP, Beurlings UP, Logarithmic UP, and Nazarovs UP which have already been well studied in the FMT domain

arXiv:2204.07048Mar 24, 20221 citation

Anisotropic uncertainty principles for metaplectic operators

Elena Cordero, Gianluca Giacchi, Edoardo Pucci

We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on $L^2(\mathbb{R}^d)$ , including degenerate cases associated with symplectic matrices whose $B$ -block has nontrivial kernel. In this setting, uncertainty phenomena are shown to be intrinsically directional and confined to an effective phase-space dimension given by $\mathrm{rank}(B)$ . First, we prove sharp Heisenberg-Pauli-Weyl type inequalities involving only the directions corresponding to $\ker(B)^\perp$ , with explicit lower bounds expressed in terms of geometric quantities associated with the underlying symplectic transformation. We also provide a complete characterization of all extremizers, which turn out to be partially Gaussian functions with free behavior along the null directions of $B$ . Building on this framework, we extend the Beurling-Hörmander theorem to the metaplectic setting, obtaining a precise polynomial-Gaussian structure for functions satisfying suitable exponential integrability conditions involving both $f$ and its metaplectic transform. Finally, we prove a Morgan-type (or Gel'fand--Shilov type) uncertainty principle for metaplectic operators, identifying a sharp threshold separating triviality from density of admissible functions and showing that this threshold is invariant under metaplectic transformations. Our results recover the classical Fourier case and free metaplectic transformations as special instances, and reveal the geometric and anisotropic nature of uncertainty principles in the presence of symplectic degeneracies.We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on $L^2(\mathbb{R}^d)$, including degenerate cases associated with symplectic matrices whose $B$-block has nontrivial kernel. In this setting, uncertainty phenomena are shown to be intrinsically directional and confined to an effective phase-space dimension given by $\mathrm{rank}(B)$. First, we prove sharp Heisenberg-Pauli-Weyl type inequalities involving only the directions corresponding to $\ker(B)^\perp$, with explicit lower bounds expressed in terms of geometric quantities associated with the underlying symplectic transformation. We also provide a complete characterization of all extremizers, which turn out to be partially Gaussian functions with free behavior along the null directions of $B$. Building on this framework, we extend the Beurling-Hörmander theorem to the metaplectic setting, obtaining a precise polynomial-Gaussian structure for functions satisfying suitable exponential integrability conditions involving both $f$ and its metaplectic transform. Finally, we prove a Morgan-type (or Gel'fand--Shilov type) uncertainty principle for metaplectic operators, identifying a sharp threshold separating triviality from density of admissible functions and showing that this threshold is invariant under metaplectic transformations. Our results recover the classical Fourier case and free metaplectic transformations as special instances, and reveal the geometric and anisotropic nature of uncertainty principles in the presence of symplectic degeneracies.

math.APCross-listedarXiv:2601.16279Jan 26, 2026

The N-Dimensional Uncertainty Principle for the Free Metaplectic Transformation

Rui Jing, Bei Liu, Rui Li +1 more

The free metaplectic transformation is an N-dimensional linear canonical transformation. This transformation operator is useful, especially for signal processing applications. In this paper, in order to characterize simultaneously local analysis of a function (or signal) and its free metaplectic transformation, we extend some different uncertainty principles (UP) from quantum mechanics including Classical Heisenberg’s uncertainty principle, Nazarov’s UP, Donoho and Stark’s UP, Hardy’s UP, Beurling’s UP, Logarithmic UP, and Entropic UP, which have already been well studied in the Fourier transform domain.The free metaplectic transformation is an N-dimensional linear canonical transformation. This transformation operator is useful, especially for signal processing applications. In this paper, in order to characterize simultaneously local analysis of a function (or signal) and its free metaplectic transformation, we extend some different uncertainty principles (UP) from quantum mechanics including Classical Heisenberg’s uncertainty principle, Nazarov’s UP, Donoho and Stark’s UP, Hardy’s UP, Beurling’s UP, Logarithmic UP, and Entropic UP, which have already been well studied in the Fourier transform domain.

MathematicsOct 1, 202028 citations

We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on

L^2(\mathbb{R}^d)

, including degenerate cases associated with symplectic matrices whose

B

-block has nontrivial kernel. In this setting, uncertainty phenomena are shown to be intrinsically directional and confined to an effective phase-space dimension given by

\mathrm{rank}(B)

. First, we prove sharp Heisenberg-Pauli-Weyl type inequalities involving only the directions corresponding to

\ker(B)^\perp

, with explicit lower bounds expressed in terms of geometric quantities associated with the underlying symplectic transformation. We also provide a complete characterization of all extremizers, which turn out to be partially Gaussian functions with free behavior along the null directions of

B

. Building on this framework, we extend the Beurling-Hörmander theorem to the metaplectic setting, obtaining a precise polynomial-Gaussian structure for functions satisfying suitable exponential integrability conditions involving both

f

and its metaplectic transform. Finally, we prove a Morgan-type (or Gel'fand--Shilov type) uncertainty principle for metaplectic operators, identifying a sharp threshold separating triviality from density of admissible functions and showing that this threshold is invariant under metaplectic transformations. Our results recover the classical Fourier case and free metaplectic transformations as special instances, and reveal the geometric and anisotropic nature of uncertainty principles in the presence of symplectic degeneracies.