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International Journal of Mathematics and Mathematical Sciences

On Some Inequalities of Uncertainty Principles Type in Quantum Calculus

Ahmed Fitouhi, Néji Bettaibi, Rym H. Bettaieb, Wafa Binous

Published Jan 1, 2008• 1 citation
Publisher

Abstract

The aim of this paper is to generalize the q‐Heisenberg uncertainty principles studied by Bettaibi et al. (2007), to state local uncertainty principles for the q‐Fourier‐cosine, the q‐Fourier‐sine, and the q‐Bessel‐Fourier transforms, then to provide an inequality of Heisenberg‐Weyl‐type for the q‐Bessel‐Fourier transform.

DOI
10.1155/2008/465909

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The aim of this paper is to generalize the q‐Heisenberg uncertainty principles studied by Bettaibi et al. (2007), to state local uncertainty principles for the q‐Fourier‐cosine, the q‐Fourier‐sine, and the q‐Bessel‐Fourier transforms, then to provide an inequality of Heisenberg‐Weyl‐type for the q‐Bessel‐Fourier transform.
Some properties of the qqq-Fourier-sine transform are studied and qqq-analogues of the Heisenberg uncertainty principle is derived for the qqq-Fourier-cosine transform studied in \cite{FB} and for the qqq-Fourier-sine transform.
A sharper uncertainty inequality which exhibits a lower bound larger than that in the classical N -dimensional Heisenberg’s uncertainty principle is obtained, and extended from N -dimensional Fourier transform domain to two N -dimensional fractional Fourier transform domains. The conditions that reach the equality relation of the uncertainty inequalities are deduced. Example and simulation are performed to illustrate that the newly derived uncertainty principles are truly sharper than the existing ones in the literature. The new proposals’ applications in time-frequency analysis and optical system analysis are also given.