- We obtain in explicit form the unique Gaussian cubature for balls (spheres) in n 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. 2003 Elsevier B.V. All rights reserved.