Extremal Systems of Points and Numerical Integration on the Sphere

This paper considers extremal systems of points on the unit sphere S ^r⊂R ^r+1, related problems of numerical integration and geometrical properties of extremal systems. Extremal systems are systems of d _n =dim P _n points, where P _n is the space of spherical polynomials of degree at most n, which maximize the determinant of an interpolation matrix. Extremal systems for S ² of degrees up to 191 (36,864 points) provide well distributed points, and are found to yield interpolatory cubature rules with positive weights. We consider the worst case cubature error in a certain Hilbert space and its relation to a generalized discrepancy. We also consider geometrical properties such as the minimal geodesic distance between points and the mesh norm. The known theoretical properties fall well short of those suggested by the numerical experiments.