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

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.