<p>In this paper, we introduce and investigate the Directional Short-Time Free Metaplectic Transform (DSTFMT), a novel time–frequency representation associated with free symplectic matrices. This framework unifies the directional short-time Fourier transform with the free metaplectic theory, encompassing several classical operators such as directional fractional Fourier transform, Fresnel, and linear canonical transforms as special cases. We establish fundamental structural properties of the DSTFMT, including its representation as a metaplectic transform of a directionally windowed function, orthogonality relations, an exact inversion formula, and a Plancherel-type identity. Furthermore, we prove a Hausdorff–Young inequality, derive a Pitt-type weighted inequality, and establish a logarithmic uncertainty principle in the metaplectic setting. These results extend several classical inequalities in harmonic analysis to a directional and symplectic time–frequency framework. The proposed transform provides a unified approach to directional phase-space analysis and offers potential applications in signal processing, quantum mechanics, and optical systems governed by symplectic structures.</p>

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The directional short-time free metaplectic transform

  • Kamel Brahim

摘要

In this paper, we introduce and investigate the Directional Short-Time Free Metaplectic Transform (DSTFMT), a novel time–frequency representation associated with free symplectic matrices. This framework unifies the directional short-time Fourier transform with the free metaplectic theory, encompassing several classical operators such as directional fractional Fourier transform, Fresnel, and linear canonical transforms as special cases. We establish fundamental structural properties of the DSTFMT, including its representation as a metaplectic transform of a directionally windowed function, orthogonality relations, an exact inversion formula, and a Plancherel-type identity. Furthermore, we prove a Hausdorff–Young inequality, derive a Pitt-type weighted inequality, and establish a logarithmic uncertainty principle in the metaplectic setting. These results extend several classical inequalities in harmonic analysis to a directional and symplectic time–frequency framework. The proposed transform provides a unified approach to directional phase-space analysis and offers potential applications in signal processing, quantum mechanics, and optical systems governed by symplectic structures.