Extremal functions of cubature formulas on a multidimensional sphere and spherical splines

We establish the general form of extremal cubature formulas on multidimensional spheres. The domains of definition for the cubature formulas under consideration are Sobolev-type spaces on the sphere. The smoothness of the class function under study may be fractional. We prove that, for a given set of nodes, there exists a one-to-one correspondence between the set of extremal functions of cubature formulas on the sphere and the set of natural spherical splines with zero spherical mean.