Spherical cubature formulas in Sobolev spaces

Abstract

We study sequences of cubature formulas on the unit sphere in a multidimensional Euclidean space. The grids for the cubature formulas under consideration embed in each other consecutively, forming in the limit a dense subset on the initial sphere. As the domain of cubature formulas, i.e. as the class of integrands, we take spherical Sobolev spaces. These spaces may have fractional smoothness. We prove that, among all possible spherical cubature formulas with given grid, there exists and is unique a formula with the least norm of the error, an optimal formula. The weights of the optimal cubature formula are shown to be solutions to a special nondegenerate system of linear equations. We prove that the errors of cubature formulas tend to zero as the number of nodes grows indefinitely.