Constructing cubature formulae for spheres and balls

It has been shown recently that cubature formulae for the unit sphere and for the unit ball are closely related; in particular, cubature formulae for the surface measure on the sphere correspond to formulae for the Chebyshev weight function on the ball. This provides a new method to generate cubature formulae on these regions. In this paper we construct a number of cubature formulae for the Chebyshev weight function on the unit ball in R2, and use them to derive new formulae for the surface measure on the sphere S2.