Extended Gaussian type cubatures for the ball

摘要

We construct cubatures that approximate the integral of a function u over the unit ball by the linear combination of surface integrals over the unit sphere of normal derivatives of u and surface integrals of u and Δ2u over m spheres, centered at the origin. We derive explicitly the weights and the nodes of these cubatures, and show that they are exact for all (2m+2)-harmonic functions.