Spherical orthogonal polynomials and symbolic-numeric Gaussian cubature formulas

It is well-known that the classical univariate orthogonal polynomials give rise to highly efficient Gaussian quadrature rules. We show how the classical orthogonal polynomials can be generalized to a multivariate setting and how this generalization leads to Gaussian cubature rules for specific families of multivariate polynomials. The multivariate homogeneous orthogonal functions that we discuss here satisfy a unique slice projection property: they project to univariate orthogonal polynomials on every one-dimensional subspace spanned by a vector from the unit hypersphere. We therefore call them spherical orthogonal polynomials.