Hardy's theorem for the q-Bessel Fourier transform

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 $f(x)=\text{const}.e^{-{1/2} x^2}$.