Quadratic-phase deformed Hankel transform: analytical framework and uncertainty principles

In this article, we study the quadratic-phase deformed Hankel transform as a natural extension of the deformed Hankel transform within the framework of linear canonical transforms, thereby unifying and generalizing several classical integral transforms. We first establish its core analytical structure, including inversion formulas, Plancherel-type identities and boundedness properties. Building on this framework, we formulate a comprehensive suite of uncertainty principles tailored to the proposed transform, encompassing the sharp logarithmic, Pitt-type, and Beckner-type inequalities, as well as concentration-based results such as the Donoho-Stark and Matolcsi-Szucs inequalities. We culminate our results by formulating the Shapiro-type uncertainty principles for orthonormal systems. These results significantly extend classical uncertainty theory to a broader setting and highlight the analytical versatility of the proposed transform.