Abstract
The aim of this article is to introduce a fractional space–time Fourier transform (FrSFT) by generalizing the fractional Fourier transform for 16-dimensional space–time \(C \hspace{-1.00006pt}\ell _{3,1}\)-valued signals over the domain of space–time (Minkowski space) \(\mathbb {R}^{3, 1}\). The primary analysis includes the investigation of fundamental properties such as the inversion, the Plancherel theorem, uniform continuity, and partial derivatives of the proposed transform. Using the space–time split, the Heisenberg uncertainty principle is established, and the FrSFT is employed to solve a partial differential equation in space–time analysis.
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Acknowledgements
The authors would like to express their sincere gratitude to the referees for their valuable and insightful reviews. Their constructive comments and suggestions have significantly contributed to enhancing the quality and clarity of the paper. The authors also extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the large group Research Project under Grant Number RGP2/237/44.
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Zayed, M., El Haoui, Y. Fractional Fourier transform for space–time algebra-valued functions. J. Pseudo-Differ. Oper. Appl. 14, 58 (2023). https://doi.org/10.1007/s11868-023-00553-3
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DOI: https://doi.org/10.1007/s11868-023-00553-3
Keywords
- Space–time algebra
- Minkowski algebra
- Space–time Fourier transform
- Fractional Fourier transform
- Uncertainty principles