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arXiv:math/0602658

Uncertainty Principles for the Fourier Transforms in Quantum Calculus

Néji Bettaibi, Ahmed Fitouhi, Wafa Binous

Published Feb 28, 2006• 1 citation
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Abstract

Some properties of the $q$-Fourier-sine transform are studied and $q$-analogues of the Heisenberg uncertainty principle is derived for the $q$-Fourier-cosine transform studied in \cite{FB} and for the $q$-Fourier-sine transform.

DOI
10.48550/arxiv.math/0602658

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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.
Abstract Uncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.
The aim of this paper is to derive the uncertainty principle which has implications in signal analysis and in quantum mechanics. First, we derive the definition of the wavelet transform in [Formula: see text]-calculus by using some weight function. Certain properties like linearity, scaling, translation, etc. were discussed. Later on, to illustrate this integral transform several results were derived for the effectiveness and performance of the proposed method. Also, this paper surveys recent applications to establish generalized uncertainty principle.