Akram Nemri

Analytical and numerical applications for the Fourier multiplier operators on ℝn × (0, ∞)

We study the Fourier multiplier operators , where and ; and we establish for them some versions of uncertainty principles. Moreover, we give an application of the general theory of reproducing kernels to the Tikhonov regularization for . Meanwhile, we give the approximate formulas for on a Hilbert space . Further, we shall establish error estimates for our approximation formulas. Finally, by using computers, we shall illustrate numerical experiments approximation formulas in .

On the connection between heat and wave problems in quantum calculus and applications

Using the q2-Laplace transform and its inverse transform introduced early on by Hahn and deeply studied by Abdi we have prove that the q-analogues of the heat and wave equations are linked as in the classical case of Bragg and Dettman. As an application, we proved first, through the q-wave polynomials, that the q-Hermite and the q-little Jacobi polynomials are related. Second, we have given a q-analogue of the Poisson kernel studied by Fitouhi and Annabi.

On harmonic analysis related with the generalized Dunkl operator

This paper deals with a new singular differential-difference operator Y t, A on the real line including, as a particular case, the Dunkl, Dunkl–Heckman and Dunkl–Cherednik operators. We establish some results of Harmonic analysis related to this operator, such that a product formula for the related eigenfunction G λ is given as integral with an explicit kernel. This product formula is an important tool to define the generalized translation operator which is used to set up a convolution structure. Next, we establish an inversion formula and prove a generalized Plancherel theorem for this operator. As a direct application, we give a maximum principle of the operators and we solve the heat equation associated with the generalized Dunkl operator on the real line.

A characterization of weighted Besov spaces in quantum calculus

Abstract In this paper, subspaces of Lp(ℝq;+) are defined using q-translations Tq,x operator and q- differences operator, called q-Besov spaces. We provide characterization of these spaces by using the q-convolution product.

Analytical approximation formulas in quantum calculus

We study the class of q-Fourier multiplier operators T m : = F q ( F q ) , which are acted on the q-Sobolev space H * , q s ( ℝ q ) , and we obtain the exact expression and some properties for the extremal functions of the best approximation problem in quantum calculus inf f ∈ H * , q s ( ℝ q ) { η ∥ f ∥ H * , q s ( ℝ q ) 2 + ∥ g − T m f ∥ L 2 ( ℝ q , + ) 2 } , where η > 0 and g ∈ L 2 ( ℝ q , + ) . As an application, we provide numerical approximate formulas for a limit case η ↑ 0 ; using q-calculus, which generalizes the Gauss-Kronrod method studied given in [14] in one-dimensional space.

Lipschitz and Besov spaces in quantum calculus

The purpose of this paper is to investigate the harmonic analysis on the time scale 𝕋q, q ∈ (0, 1) to introduce q-weighted Besov spaces subspaces of Lp(𝕋 q) generalizing the classical one. Further, using an example of q-weighted wα,β(.; q) which is introduced and studied. We give a new characterization of the q-Besov space using q-Poisson kernel and the g1 Littlewood–Paley operator.

uncertainty principles for the Fourier transform with numerical aspect

In this paper, we establish local uncertainty principle for the Fourier transform; and we deduce version of Heisenberg–Pauli–Weyl uncertainty principle. We use also the local uncertainty principle, the partial Fourier integrals and the techniques of Donoho–Stark, we present two uncertainty principles of concentration type in the theory, when . Some numerical applications are given.

POLYNOMIALS EXPANSIONS FOR SOLUTION OF WAVE BESSEL EQUATION IN QUANTUM CALCULUS

In this paper, using the q2-Laplace transform early introduced by Abdi,1 we study q-wave Bessel polynomials related with the q-Bessel operator Δq,α. We show in particular that they are linked to the q-little Jacobi polynomials pn(x;α, β|q2).