Abstract
We obtain in explicit form the unique Gaussian cubature for balls (spheres) in R 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 2 m−1. A Gaussian cubature for simplices is also constructed. Upper bounds for the errors for certain smoothness classes are derived.
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