${\mathcal{O}}_{\alpha }$-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 $\alpha \mathrm{\notin }\pi \mathbb{Z}$. 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{ö}rmander’s theorem.