Quadratic-Phase Fourier--Bessel Transform: definitions, properties and uncertainty principles

In this manuscript, we introduce the quadratic–phase Fourier–Bessel transform and develop its foundational properties, including continuity, the Riemann–Lebesgue lemma, reversibility, and Parseval’s identity. We define the associated translation operator and convolution product, establishing their main properties within this framework. As an application, we prove a Donoho-Stark-type uncertainty principle for the quadratic-phase Fourier–Bessel transform, extending classical uncertainty results to this generalized setting.