Papers by Néjib Ben Salem
Positivity, May 7, 2010
We define Hilbert transform and conjugate Poisson integrals associated with the Jacobi differenti... more We define Hilbert transform and conjugate Poisson integrals associated with the Jacobi differential operator on (0, +∞). We prove that these operators are bounded in the appropriate Lebesgue spaces L p , 1 < p < +∞. In this study, the tools used are the Littlewood-Paley g-functions associated with the Poisson semigroup and the supplementary Poisson semigroup which we introduce in this paper. Keywords Jacobi operator • Jacobi function • Hilbert transform • Littlewood function • Conjugate Poisson integral Mathematics Subject Classification (2000) 44A15 • 43A15 • 43A32 where A α,β (x) = 2 2ρ (sinh x) 2α+1 (cosh x) 2β+1 , ρ = α + β + 1.
Hilbert transforms and related topics associated with the Dunkl-Hermite functions
Integro-di�erential-di�erence equations associated with the Dunkl operator and entire functions
Fractional Calculus and Applied Analysis
Using a convolution structure on the real line associated with the Jacobi-Dunkl differential-diff... more Using a convolution structure on the real line associated with the Jacobi-Dunkl differential-difference operator Λ α,β given by: Λ α,β f (x) = f (x) + ((2α + 1) coth x + (2β + 1) tanh x) f (x) − f (−x) 2 , α ≥ β ≥ − 1 2 , we define mean-periodic functions associated with Λ α,β. We characterize these functions as an expansion series intervening appropriate elementary functions expressed in terms of the derivatives of the eigenfunction of Λ α,β. Next, we deal with the Pompeiu type problem and convolution equations for this operator.
The Ramanujan Journal, 2006
In this paper, a product formula for the eigenfunction of the Jacobi-Dunkl differential-differenc... more In this paper, a product formula for the eigenfunction of the Jacobi-Dunkl differential-difference operator is derived. It leads to a uniformly bounded convolution of point measures and a signed hypergroup on IR.
Journal of Mathematical Analysis and Applications, 1997
We prove a Mehler representation for Jacobi functions t with respect to the dual variable. We exp... more We prove a Mehler representation for Jacobi functions t with respect to the dual variable. We exploit this representation to define a pair of dual integral transforms and its transposed t. We define two second order difference ␣ ,  ␣ ,  Ž ␣ , . Ž. operators P and Q such that t is an eigenfunction of P with respect ␣,  ␣ ,  to the dual variable , and and t are permutation operators between P ␣ ,  ␣ ,  ␣ ,  and Q. Next we give some spaces of functions on which and t are ␣ ,  ␣ ,  isomorphisms and we establish inversion formulas for these transforms.
Orthogonality and distributional weight functions for Dunkl–Hermite polynomials
Integral Transforms and Special Functions, 2009
Sobolev type spaces associated with jacobi differential operators
Integral Transforms and Special Functions, 2000
Sobolev type spaces , associated with the Jacobi differential operators are studied. Some propert... more Sobolev type spaces , associated with the Jacobi differential operators are studied. Some properties including completeness and inclusion are proved. Next, the Jacobi potential is defined as a pseudodifferential operator associated with a precise symbol. The operator is extended to a space of distributions. The L -space of all such Jacobi potential is defined. It is proved that this space
Analytic mean-periodic functions associated with the Dunkl operator in a disk
Complex Variables, Theory and Application: An International Journal, 2005
ABSTRACT
Sobolev type spaces associated with the Jacobi-Dunkl operator
FRACTIONAL CALCULUS AND APPLIED …, 2004
In this paper, we consider the Jacobi-Dunkl operator Λα, β. We establish some results of Harmonic... more In this paper, we consider the Jacobi-Dunkl operator Λα, β. We establish some results of Harmonic analysis related to this operator which permit to introduce Sobolev type spaces associated with this operator. Next, we define the Jacobi-Dunkl potential of order s ...
Almost-Periodic Functions and Distributions Associated with the Dunkl Operator on R
Communications in Mathematical Analysis, 2009
Analysis in Theory and Applications
In this paper, we consider the generalized translations associated with the Dunkl and the Jacobi-... more In this paper, we consider the generalized translations associated with the Dunkl and the Jacobi-Dunkl differential-difference operators on the real line which provide the structure of signed hrpergroups on R. Especially, we study the representation of the generalized translations of the product of two functions for these signed hypergroups.
Shannon, Sobolev and uncertainty inequalities for the Weinstein transform
Integral Transforms and Special Functions
On a mean value property associated with the Weinstein operator
De Gruyter eBooks, Oct 13, 2011
Pizzetti Series and Polyharmonicity Associated with the Dunkl Laplacian
Mediterranean Journal of Mathematics, 2010
In this paper we are concerned with the Pizzetti series associated with the Dunkl Laplacian denot... more In this paper we are concerned with the Pizzetti series associated with the Dunkl Laplacian denoted Δ k . We study the convergence of this series and we give some applications. Next we establish some properties of polyharmonic functions associated with Δ k , especially, we establish Liouville type results.
Inequalities related to spherical harmonics associated with the Weinstein operator
Integral Transforms and Special Functions
Spherical harmonics and applications associated with the Weinstein operator
Potential Theory - ICPT 94, 1996
This paper consists in a first study of the Hardy space H 1 in the rational Dunkl setting. Follow... more This paper consists in a first study of the Hardy space H 1 in the rational Dunkl setting. Following Uchiyama's approach, we characterizee H 1 atomically and by means of the heat maximal operator. We also obtain a Fourier multiplier theorem for H 1. These results are proved here in the one-dimensional case and in the product case.
Abstract. In this work we consider the Dunkl operator on the complex plane, defined by
Shapiro Type Inequalities for the Weinstein and the Weinstein-Gabor Transforms
The aim of this paper is to prove new uncertainty principles for the Weinstein and the Weinstein-... more The aim of this paper is to prove new uncertainty principles for the Weinstein and the Weinstein-Gabor transforms associated with the Weinstein operator dened on the half space $\mathbb{R}^d_{+}$ by $\Delta_W =\sum_{i=1}^{d } \frac{\partial}{\partial x_i^2}+ \frac{2\alpha+1}{x_{d}}\frac{\partial}{\partial x_{d-1}};\ \ \ \ \ d\ge2,\ \alpha>-1/2.$ More precisely, we give a Shapiro-type uncertainty inequality for the Weinstein transform that is, for $s>0$ and $\{\phi_n\}_n$ be an orthonormal sequence in $L^2_\alpha(\mathbb{R}^d_{+})$, $\sum_{n=1}^N(\Vert \vert x\vert^s \phi_n\Vert_{{L_\alpha^2(\mathbb{R}^d_{+})}}^{2}+ \Vert \vert\xi\vert^s \mathcal{F}_W(\phi_n)\Vert_{{L_\alpha^2(\mathbb{R}^d_{+})}}^{2 })\geq KN^{1+\frac{s}{2\alpha+d+1}},$ where $K$ is a constant which depends only on $d$; $s$ and $\alpha$. Next, we establish an analogous inequality for the Weinstein-Gabor transform
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Papers by Néjib Ben Salem