Optimal numerical integration on a sphere

Little has been published on this subject or on its extension to the solid sphere. The literature is surveyed briefly in Section 7. Most of our space is devoted to formulas invariant with respect to a finite group of rotations of the sphere. We study such formulas by means of the group characters, as does Sobolev 12, 13. The criterion by which integration formulas are usually judged is that of efficiency. It is defined like this. Consider a system of functions over the domain of integration such as polynomials in Euclidean space or surface harmonics on the sphere. They have properties of completeness and they are ordered in a natural way. Suppose that the integration formula is exact for the first L independent functions and therefore for all linear combinations of them. The efficiency E is the ratio of L to the number of arbitrary constants in the formula. The latter is a fixed multiple (one more than the dimensionality of the domain of integration) of the number N of points at which the integrand is evaluated. A linear combination of surface harmonics (of degree not more than p) will be called a spherical polynomial (of degree p). If we choose to embed the surface of the sphere in Euclidean space of three dimensions, we find that the trace left on the surface by an ordinary polynomial in x, y and z is a spherical polynomial of the same degree. For the surface of the sphere a pth degree integration formula (exact for spherical polynomials of degree p) has