Cubature Formulas on Symmetric Homogeneous Spaces

The problem of numerical integration over the surface of the unit sphere is one of the oldest and mostimportant in Analysis.

The theory of functions on the sphere has been initiated in the XVIIIth century in works of Laplace and Legendre when the first cubature formulas appeared.

We give an explicit solution of a well-known problem on optimal cubature formulas on sets of differentiable functions on the unit sphere which remained open for a long time.