Quadratic-Phase Fourier--Bessel Transform: definitions, properties and uncertainty principles
Ahmed Saoudi
TL;DR
The paper develops the quadratic-phase Fourier--Bessel transform $\mathbb{B}^{a,b,c}_{d,e,\gamma}$ on the half-line, introducing its kernel $\mathbb{J}^{a,b,c}_{d,e,\gamma}$ and showing that with $\gamma>-1/2$ and $b\neq 0$ this transform generalizes several classical transforms via parameter choices. It builds a complete harmonic-analytic framework by proving continuity, the Riemann–Lebesgue lemma, invertibility with $\mathbb{B}^{-c,-b,-a}_{-e,-d,\gamma}$, and Parseval/Plancherel identities, then defines a translation operator and a convolution compatible with the transform, establishing boundedness and Young-type estimates. The main application is a Donoho–Stark-type uncertainty principle for the quadratic-phase Fourier--Bessel domain, including time–frequency concentration bounds and a concentration-type inequality that recovers the classical case when $p=q=2$. These results lay a rigorous foundation for time-frequency analysis in quadratic-phase Fourier–Bessel settings and pave the way for further developments such as wavelet theory and Calderón reproducing formulas in this generalized Bessel framework.
Abstract
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.