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

TL;DR

This paper introduces the quadratic-phase Fourier--Bessel transform, explores its fundamental properties, and establishes an uncertainty principle, expanding the mathematical framework for signal analysis with this new transform.

Contribution

It presents the first comprehensive development of the quadratic-phase Fourier--Bessel transform, including properties, operators, and an uncertainty principle, advancing theoretical understanding in this area.

Findings

Established continuity, reversibility, and Parseval's identity for the transform.

Defined translation and convolution operators with key properties.

Proved a Donoho-Stark-type uncertainty principle for the transform.

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.