Some Cubature Formulas for the Surface of the n-Sphere

Abstract

Let $U_n $ denote the surface of the unit n-sphere. Given a cubature formula of degree d for $U_{n - 1} $ with weight function unity, we give a method for constructing cubature formulas of degree d for $U_n $ with weight function $w(x_1 )$. The specific case in which $w(x_1 )$ is the Poisson kernel is discussed, and some new fifth-degree formulas are obtained. The resulting formulas can be considered as generalizations of the spherical product Gauss and spherical product Lobatto cubature formulas for $U_n $. It is shown that a harmonic interpolation formula for the unit n-sphere is equivalent to a cubature formula for $U_n $ with $w(x_1 )$ the Poisson kernel.