Heisenberg Uncertainty Principle for the q-Bessel Fourier transform

TL;DR

This paper establishes an uncertainty inequality for the q-Bessel Fourier transform using an entropy-based argument, extending the classical Heisenberg uncertainty principle to a q-analogue involving q-Bessel functions.

Contribution

It introduces a novel uncertainty inequality for the q-Bessel Fourier transform using an Hirschman-Beckner entropy approach, expanding the theoretical framework of Fourier analysis in q-calculus.

Findings

Derived a new uncertainty inequality for the q-Bessel Fourier transform.

Extended classical Fourier uncertainty principles to the q-analogue setting.

Utilized entropy methods to establish the inequality.

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.