Heisenberg Uncertainty Principle for the q-Bessel Fourier transform

Abstract

In this paper we uses an I.I. Hirschman-W. Beckner entropy argument to give an uncertainty inequality for the $q$-Bessel Fourier transform: $${\mathcal{F}}_{q,v}f(x)=c_{q,v}{\int }_0^{\infty }f(t)j_v(xt,q^2)t^{2v+1}{d}_{q}t,$$ where $j_v(x,q)$ is the normalized Hahn-Exton $q$-Bessel function.