Cubature formulae for spheres, simplices and balls

We obtain in explicit form the unique Gaussian cubature for balls (spheres) in Rn based on integrals over balls (spheres), centered at the origin, that integrates exactly all m-harmonic functions. In particular, this formula is exact for all polynomials in n variables of degree 2m−1. A Gaussian cubature for simplices is also constructed. Upper bounds for the errors for certain smoothness classes are derived.