Journal ArticleDOI
Pizzetti Series and Polyharmonicity Associated with the Dunkl Laplacian
Néjib Ben Salem,Kamel Touahri +1 more
TLDR
In this paper, the convergence of the Pizzetti series associated with the Dunkl Laplacian is studied and some properties of polyharmonic functions associated with this series are established.Abstract:
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.read more
Citations
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Journal ArticleDOI
On mean-value properties for the Dunkl polyharmonic functions
TL;DR: In this paper, the authors derive differential relations between the spherical and solid means of continuous functions and use them to give inductive proofs of mean-value properties for the polyharmonic functions and their converses.
Journal ArticleDOI
Cubature Formulae Associated with the Dunkl Laplacian
Néjib Ben Salem,Kamel Touahri +1 more
TL;DR: In this paper, the authors studied the integration of functions of the form ∆ ∆ = ∆ + ∆, ∆ − ∆ with respect to the Dunkl Laplacian Δk and gave cubature formulae having highest order of precision for functions of degree m. In particular, they gave an extension of the Pizzetti formula type for functions in ∆ 2m-1.
Posted Content
An extension of Pizzetti's formula associated with the Dunkl operators
Nobukazu Shimeno,Naoya Tani +1 more
TL;DR: In this paper, an extension of Pizzetti's formula associated with the Dunkl operators is given, where the inner product of an arbitrary function and a homogeneous Dunkl harmonic polynomial on the unit sphere is given.
Journal Article
A Liouville theorem for polyharmonic functions
TL;DR: In this paper, it was shown that if a polyharmonic function on R is a polynomial function, and the growth of the function is suitably restricted, then it must be a polyphonic function.
Book ChapterDOI
An extension of Pizzetti's formula associated with the Dunkl operators
Nobukazu Shimeno,Naoya Tani +1 more
TL;DR: In this paper, an extension of Pizzetti's formula associated with the Dunkl operators is given, where the inner product of an arbitrary function and a homogeneous Dunkl harmonic polynomial on the unit sphere is given.
References
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Journal ArticleDOI
Encyclopedia of Mathematics and its Applications.
William B. Jones,W. J. Thron +1 more
Journal ArticleDOI
Differential-difference operators associated to reflection groups
TL;DR: In this article, a theory of spherical harmonics for measures invariant under a finite reflection group is presented, where the measures are products of powers of linear functions, whose zero-sets are the mirrors of the reflections in the group, times the rotation-invariant measure on the unit sphere in Rn.
Book
Orthogonal Polynomials of Several Variables
Charles F. Dunkl,Yuan Xu +1 more
TL;DR: In this article, the authors considered the properties of orthogonal polynomials on the unit sphere, root systems and Coxeter groups, and the Summability of Orthogonal expansions.
Journal ArticleDOI
Generalized hermite polynomials and the heat equation for dunkl operators
TL;DR: Based on the theory of Dunkl operators, the authors presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on ℝ N fixme.