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An uncertainty principle for the basic Bessel transform

Authors:

Ahmed Fitouhi Néji Bettaibi Wafa Binous Hédi Ben Elmonser

Institution:

(1) Faculté des Sciences de Tunis, 1060 Tunis, Tunisia;(2) Institut Préparatoire aux études d’Ingénieur de Nabeul, 8000 Nabeul, Tunisia;(3) Institut Biotechnologie de Béja, 9000 Béja, Tunisia;(4) Institut Préparatoire aux études d’Ingénieur de Bizerte, Bizerte, Tunisia

Abstract:

The aim of this paper is to prove an uncertainty principle for the basic Bessel transform of order $\alpha \geq -\frac{1}{2}$ . In order to obtain a sharp uncertainty principle, we introduce and study a generalized q-Bessel-Dunkl transform which is based on the q-eigenfunctions of the q-Dunkl operator newly given by:

$T_{\alpha,q}(f)(x)=D_{q}f(x)+\frac{2\alpha +1]_{q}}{2q^{2\alpha +1}}\frac{f(x)-f(-x)}{x}.$

In this work, we will follow the same steps of Fitouhi et al. (Math. Sci. Res. J., 2007) using the operator T _α,q instead of the q-derivative.

Keywords:

Quantum calculus q-Jackson integrals q-Bessel function q-Bessel transform q-Dunkl operator

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