${\mathcal{O}}_{\hat{I}}\pm$-transformation and its uncertainty principles

Abstract

In this paper, we introduce a family of integral transforms, denoted by \(\mathcal{O}_α\), and constructed via kernel fusion of the fractional Fourier transform (FRFT) with angle \(α\notin π\mathbb{Z}\). We demonstrate that the \(\mathcal{O}_α\)-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 ${\mathcal{O}}_{\hat{I}}\pm$-transform, including Heisenberg's inequality, logarithmic uncertainty inequality, local uncertainty inequality, Hardy's inequality, Pitt's inequality, and Beurling-H{ö}rmander's theorem.

13 pages, accepted by Integral Transforms Spec. Funct