Hardy's theorem for the q-Bessel Fourier transform

TL;DR

This paper establishes a q-analogue of Hardy's theorem specifically for the q-Bessel Fourier transform, characterizing functions with Gaussian decay in this context.

Contribution

It introduces a novel q-analogue of Hardy's theorem tailored for the q-Bessel Fourier transform, extending classical Fourier analysis results.

Findings

Proves the q-analogue of Hardy's theorem for the q-Bessel Fourier transform.

Characterizes functions with Gaussian decay in the q-Bessel Fourier setting.

Abstract

In this paper we give a q-analogue of the Hardy's theorem for the $q$-Bessel Fourier transform. The celebrated theorem asserts that if a function $f$ and its Fourier transform $\hat{f}$ satisfying $|f(x)|\leq c.e^{-{1/2} x^2}$ and $|\hat{f}(x)|\leq c.e^{-{1/2} x^2}$ for all $x\in\mathbb{% R}$ then $f(x)=\text{const}.e^{-{1/2} x^2}$.