Search

In this paper, utilizing the Dunkl transform and a generalized translation operator, we derive an analogue of the Riesz-Feller fractional derivative for a class of functions satisfying Lipschitz conditions. [ABSTRACT FROM AUTHOR]

We introduce in this paper the concept of Dunkl-coherent pair of forms (linear functionals) in the symmetric case. We prove that if { u , v } is a Dunkl-symmetrically coherent pair of symmetric forms, then at least one of them has to be a symmetric Dunkl-classical form, i.e. Generalized Hermite or Generalized Gegenbauer polynomials. [ABSTRACT FROM AUTHOR]

We consider the Dunkl differential-difference operator associated with a reflection group W and a nonnegative parameter function k. We exhibit and study a new family of sharp integral inequalities related to the Dunkl heat semigroup generated by the Dunkl-Laplacian . These estimations are a natural extension and refinement of the Poincaré inequality established in [1] and used to derive other applications. [ABSTRACT FROM AUTHOR]

We generalize classical Hobson's formula concerning partial derivatives of radial functions on a Euclidean space to a formula in the Dunkl analysis. As applications we give new simple proofs of known results involving Maxwell's representation of harmonic polynomials, Bochner-Hecke identity, Pizzetti formula for spherical mean, and Rodrigues formula for generalized Hermite polynomials. [ABSTRACT FROM AUTHOR]

We study the boundedness of the multiplier of the intervalfor the Dunkl transform of orderon weightedspaces, with. In particular, we get that it is bounded frominto itself if and only ifwhenor if and only ifwhen. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new function spaces that are denoted by,, and that are spaces oftype where the Dunkl convolution and the Dunkl transformation are defined. Some properties of these spaces are studied. We study the convolutors and the surjective Dunkl convolution operator acting on. In the case, we obtain complete characterizations. [ABSTRACT FROM AUTHOR]

In this work, we establish a Babenko–Beckner-type inequality for the Dunkl transform of a radial function and Dunkl transform associated to a finite reflection group generated by the sign changes. As applications, we establish a new version of Young’s-type inequality and uncertainty relation for Renyi entropy. [ABSTRACT FROM PUBLISHER]

The aim of this work is to study the symmetric orthogonal polynomials’ sequences which are Dunkl-classical. We recover again that the generalized Hermite polynomials and the generalized Gegenbauer polynomials are the only symmetric Dunkl-classical through a different manner that in Ben Cheikh and Gaied [Characterization of the Dunkl-classical symmetric orthogonal polynomials. Appl Math Comput. 2007;187:105–114] by solving a differential-difference equation in the dual space of polynomials. Moreover, we deduce the structure relation that such a symmetric monic orthogonal polynomials satisfies. [ABSTRACT FROM AUTHOR]

We present a new characterization of Dunkl-classical symmetric orthogonal polynomials. Moreover, we establish an example of a non-symmetric sequence of Dunkl-classical orthogonal polynomials, which proved that the set of this last kind of polynomials is not empty. [ABSTRACT FROM AUTHOR]

Using a generalized spherical mean operator, we obtain an analog of Titchmarsh's Theorem for the Dunkl transform for functions satisfying the spherical Lipschitz condition in L2(d, ωk), where ωk is a weight function invariant under the action of an associated Weyl group. [ABSTRACT FROM AUTHOR]

On the real line, the Dunkl operators are differential-difference operators associated with the reflection group 2 on . In this paper, we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator associated with the Dunkl operator on  from the spaces Lp,α() to the spaces Lq,α() and from the spaces L1,α() to the weak spaces WLq,α(). [ABSTRACT FROM AUTHOR]

We establish a new expression of the generalized Poisson transform associated with Dunkl operator using the mean value operator, and we give several applications related to the generalized Poisson equation. [ABSTRACT FROM AUTHOR]

In this work, we give a new proof for the orthogonality of the Dunkl-Hermite polynomials [image omitted] , when the index μ>-1/2, via quasi-monomiality techniques and show that when-m-1/2<μ<-m+1/2, where m is a positive integer, [image omitted] is an orthogonal set with respect to a sesquilinear form which leads to a Krein space structure. Then, given a family of polynomials orthogonal with respect to a linear functional ωμ expressed in terms of iteratives of the Dunkl operator on the real line, we treat the problem of extending ωμ to a space of test functions which includes polynomials. As an example, we treat the extension of the weight function with respect to which the Dunkl-Hermite polynomials are orthogonal. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the harmonic analysis associated with Dunkl's operator on the real line to introduce a Besov type spaces generalizing the classical one. Different characterizations of these spaces are given using generalized translation operator and also by mean modulus of continuity. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the Dunkl-Appell d-orthogonal polynomials. Various properties of the obtained polynomials are singled out: a generating function, a (d+1)-order recurrence relation and a differential-difference equation. We also explicitly express the d-dimensional functional for which the d-orthogonality holds. A detailed discussion of the (d+1)-symmetric solutions is presented. Some relationships between the obtained polynomials and some known orthogonal polynomial families are established. [ABSTRACT FROM AUTHOR]

In this work, by using the Wiener-Tauberian Theorem, we establish a Pompeiu-type theorem (which consists to study the uniqueness of some convolution equations solutions) associated with a differential-difference operator ?? a , called the Dunkl operator on the real line. Next, we give some applications. [ABSTRACT FROM AUTHOR]

In this article, we consider the Dunkl operator of index k , k ≥ 0, on . We introduce analytic mean-periodic functions associated with in a disk and we study the expansion of a mean-periodic function in series with respect to appropriate exponential monomials. [ABSTRACT FROM AUTHOR]

In this work, we consider the Dunkl operator on the real line ${I\hskip-3pt R}$ , defined for $f\in C^{1}\lpar{I\hskip-3pt R}\rpar $ by $${\cal D}_{k}f\lpar x\rpar = f^{\prime}\lpar x\rpar + k {f\lpar x\rpar - f\lpar {-}x\rpar \over x}\comma \quad k\geq 0.$$ A C ∞ -function f on ${I\hskip-3pt R}$ , is said to be mean-periodic function associated with the Dunkl operator &Dscr; k if there exists a nonzero distribution μ on ${I\hskip-3pt R}$ with compact support such that $$\mu \,*_{k}\,f = 0,$$ where * k denotes the convolution associated with the Dunkl operator &Dscr; k . We characterize the solutions of these equations and we study the expansion of a mean-periodic function in series with respect to appropriate exponential monomials. Finally, we give some applications. [ABSTRACT FROM AUTHOR]