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

  Published:1997-02-01
 Printed: Feb 1997
  • Yuan Xu
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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 $|x_1|^{\alpha_1}\cdots |x_d|^{\alpha_d}$ on the unit sphere $S^{d-1}$ in $\RR^d$. The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator.
Keywords: Orthogonal polynomials in several variables, sphere, h-harmonics Orthogonal polynomials in several variables, sphere, h-harmonics
MSC Classifications: 33C50, 33C45, 42C10 show english descriptions Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]
Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
33C50 - Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
33C45 - Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]
42C10 - Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
 

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