Orthogonal Polynomials and Cubature Formulae on Spheres and on Simplices
نویسندگان
چکیده
Orthogonal polynomials on the standard simplex Σ in R are shown to be related to the spherical orthogonal polynomials on the unit sphere S in R that are invariant under the group Z2×· · ·×Z2. For a large class of measures on S cubature formulae invariant under Z2 × · · · × Z2 are shown to be characterized by cubature formulae on Σ. Moreover, it is also shown that there is a correspondence between orthogonal polynomials and cubature formulae on Σ and those invariant on the unit ball B in R. The results provide a new approach to study orthogonal polynomials and cubature formulae on spheres and on simplices.
منابع مشابه
Orthogonal Polynomials and Cubature Formulae on Spheres and on Balls∗
Orthogonal polynomials on the unit sphere in Rd+1 and on the unit ball in Rd are shown to be closely related to each other for symmetric weight functions. Furthermore, it is shown that a large class of cubature formulae on the unit sphere can be derived from those on the unit ball and vice versa. The results provide a new approach to study orthogonal polynomials and cubature formulae on spheres.
متن کاملCubature formulae for spheres, simplices and balls
We obtain in explicit form the unique Gaussian cubature for balls (spheres) in Rn based on integrals over balls (spheres), centered at the origin, that integrates exactly all m-harmonic functions. In particular, this formula is exact for all polynomials in n variables of degree 2m − 1. A Gaussian cubature for simplices is also constructed. Upper bounds for the errors for certain smoothness clas...
متن کاملNear-minimal cubature formulae on the disk
The construction of (near-)minimal cubature formulae on the disk is still a complicated subject on which many results have been published.We restrict ourselves to the case of radial weight functions and make use of a recent connection between cubature and the concept of multivariate spherical orthogonal polynomials to derive a new system of equations defining the nodes and weights of (near-)min...
متن کاملInvariant Cubature Formulae for Spheres and Balls
Invariant cubature formulae for a class of weight functions on the simplex T d are derived using combinatorial methods, extending the formulae in [Grundmann and Möller, SIAM J. Numer Anal., 15 (1978), pp. 282–290] for the unit weight function on T d. These formulae are used to derive cubature formulae on the surface of the sphere Sd and on the unit ball Bd using connections between cubature for...
متن کاملInvariant Cubature Formulae for Spheres and Balls by Combinatorial Methods
Invariant cubature formulae for a class of weight functions on the simplex T d are derived using combinatorial methods, extending the formulae in [Grundmann and Möller, SIAM J. Numer Anal., 15 (1978), pp. 282–290] for the unit weight function on T . These formulae are used to derive cubature formulae on the surface of the sphere S and on the unit ball B using connections between cubature formul...
متن کامل