An uncertainty principle for the basic Bessel transform

Abstract

The aim of this paper is to prove an uncertainty principle for the basic Bessel transform of order α ≥ − 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: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.

Heisenberg Uncertainty Principle for the q-Bessel Fourier transform

“…Note that other versions of the Heisenberg uncertainty principle for the q-Fourier transform have recently appeared in the literature [1,2,6]. There are some differences of the results cited above and our result:…”

“…• In [6] the uncertainty inequality is established for functions in q-Schwartz space. In this paper the uncertainty inequality is established for functions in L q,2,v space.…”

A Quantum Wavelet Uncertainty Principle

“…In [74], a Heisenberg-Weyl type uncertainty principle has been shown for a q-Dunkl transform. Additionally, a sharp q-Bessel-Dunkl uncertainty principle has been proved in [75], generalizing the case of the basic Bessel transform. In [76], the q-Dunkl transform on the real line has been applied to derive an L p -version of the Hardy uncertainty principle.…”