$\mathcal{O}_α$-transformation and its uncertainty principles
Lai Tien Minh, Trinh Tuan
TL;DR
The paper defines the $\mathcal{O}_{\alpha}$-transformation as a kernel fusion of the fractional Fourier transform for $α \notin π\mathbb{Z}$ and develops its foundational operator theory, including mapping properties, Parseval-type identities, and $L^p$ bounds. It then derives a comprehensive set of uncertainty principles for $\mathcal{O}_{\alpha}$, including a Heisenberg-type bound with Gaussian-equality cases, a Beckner-type logarithmic inequality, and local concentration results, complemented by Hardy-type decay and Beurling– Hörmander rigidity results. The results connect the new transform to classical Fourier analysis via the auxiliary function $\hat{f}(t)=e^{i a t^{2}} f(t)$ and $F[\hat{f}]$, yielding explicit localization limits in time–frequency settings. Overall, the work extends fractional-transform analysis by establishing robust, quantitative uncertainty principles for $\mathcal{O}_{\alpha}$ and laying the groundwork for potential time–frequency signal processing applications.
Abstract
In this paper, we introduce a family of $\mathcal{O}_α$-transformation based on kernels fusion of the fractional Fourier transform (abbreviated as FRFT) with angle $α\notin π\mathbb{Z}$. 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 $\mathcal{O}_α$-transform, including Heisenberg's inequality, logarithmic uncertainty inequality, local uncertainty inequality, Hardy's inequality, and Beurling-H{ö}rmander's theorem.