Extended Gaussian type cubatures for the ball

We construct cubatures that approximate the integral of a function uu over the unit ball by the linear combination of surface integrals over the unit sphere of normal derivatives of uu and surface integrals of uu and Δ2uΔ2u over mm 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)(2m+2)-harmonic functions.