Abstract
The linear canonical transform (LCT) is a type of integral transform that is widely used as an analytical tool in fields, such as applied mathematics, signal processing, optics, quantum physics, and filter design. In this article, we introduce an extension of the linear canonical transform to functions in the spacetime geometric algebra, termed as the spacetime linear canonical transform. We achieve this by utilizing the Fourier transformation framework in space-time algebra, while taking a novel hybrid Fourier–LCT kernel. The proposed transform provides a flexible representation of information in the spacetime linear canonical domain, which lies between the Minkowski spacetime and the spacetime Fourier frequency domain. We investigate the fundamental properties of the proposed transform, including the Plancherel theorem, uniform continuity, and partial derivatives. Furthermore, we establish several uncertainty inequalities using the observer-related spacetime split, including Heisenberg’s uncertainty principle, Hardy’s theorem, and the Hausdorff–Young’s inequality. The spacetime linear canonical transform offers a novel approach for the analysis of spacetime algebra-valued signals and holds significant potential for future research and applications in diverse fields.
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Teali, A.A. Spacetime Linear Canonical Transform and the Uncertainty Principles. Mediterr. J. Math. 20, 301 (2023). https://doi.org/10.1007/s00009-023-02502-2
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DOI: https://doi.org/10.1007/s00009-023-02502-2
Keywords
- Spacetime algebra
- spacetime Fourier transform
- spacetime linear canonical transform
- uncertainty principles