Orthogonal Polynomials and Cubature Formulae on Spheres and on Balls

Orthogonal polynomials on the unit sphere in RRd+1 and on the unit ball in RRd 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.