Orthogonal Polynomials for a Family of Product Weight Functions on the Spheres

Abstract

ABSTRACT. Based on the theory of spherical harmonics for measures invariant under a finite reflection group developed by Dunkl recently, we study orthogonal polynomials with respect to the weight functions jx 1 j ã 1 Ð Ð Ð jx d j ã d on the unit sphere S d 1 in R d . The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator.

Generalized Ellipsoidal and Sphero-Conal Harmonics

“…The contents of the present paper are also related to the book by Dunkl and Xu [4], and the papers by Liamba [14] and Xu [21], however, these works do not involve Stieltjes polynomials. We also refer to papers by Kalnins and Miller [8,9,10].…”

External Ellipsoidal Harmonics for the Dunkl-Laplacian

“…Following the book [9] by Dunkl and Xu and the paper [23] by Xu we will assume that α j ≥ 0 for each j = 0, 1, . .…”

Orthogonal Polynomials and Cubature Formulae on Spheres and on Balls

“…(It should be mentioned that the spherical coordinates adopted above are in the reverse order of the spherical coordinates used in [24].) On the other hand, an orthonormal basis for W Σ can be given in terms of the Jacobi polynomials P (α,β) n by (cf.…”

“…In an effort to understand Dunkl's theory of h-harmonics, in [24] we study the orthogonal polynomials on S d associated to H(y) = H α y…”

“…For d = 2, using the spherical coordinates y 1 = r cos θ 1 , y 2 = r sin θ 1 cos θ 2 , y 3 = r sin θ 1 sin θ 2 , r = |y|, some of the h-harmonics of degree n (Y n,(1) k in the notation of [24]) are given by…”