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arXiv · 2503.15132

$\mathcal{O}_{\alpha}$-transformation and its uncertainty principles

Lai Tien Minh·Trinh Tuan
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Abstract

In this paper, we introduce a family of integral transforms, denoted by \(\mathcal{O}_{\alpha}\), and constructed via kernel fusion of the fractional Fourier transform (FRFT) with angle \(\alpha \notin \pi \mathbb{Z}\). We demonstrate that the \(\mathcal{O}_{\alpha}\)-transformation constitutes a well-defined integral operator by establishing its basic operational properties. Besides, we survey various mathematical aspects of the uncertainty principles for the $\mathcal{O}_{\alpha}$-transform, including Heisenberg's inequality, logarithmic uncertainty inequality, local uncertainty inequality, Hardy's inequality, Pitt's inequality, and Beurling-H{\"o}rmander's theorem.

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Lai Tien Minh, Trinh Tuan. 2025-03-19. $\mathcal{O}_{\alpha}$-transformation and its uncertainty principles. https://doi.org/10.1080/10652469.2026.2635619

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