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

Abstract

In this paper, we introduce a family of $\mathcal{O}_{\alpha}$-transformation based on kernels fusion of the fractional Fourier transform (abbreviated as FRFT) with angle $\alpha \notin \pi \mathbb{Z}$. We point out this is a valid integral transform via 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, and Beurling-H{\"o}rmander's theorem.