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Quadrature formulas for spheres, the rotation group, and other compact,homogeneous manifolds are important in a number of applications and have beenthe subject of recent research. The main purpose of this paper is to studycoordinate independent quadrature (or cubature) formulas associated withcertain classes of positive definite and conditionally positive definitekernels that are invariant under the group action of the homogeneous manifold.In particular, we show that these formulas are accurate -- optimally so in manycases -- and stable under an increasing number of nodes and in the presence ofnoise, provided the set X of quadrature nodes is quasi-uniform. The stabilityresults are new in all cases. In addition, we may use these quadrature formulasto obtain similar formulas for manifolds diffeomorphic to $\mathbb{S}^n$,oblate spheroids for instance. The weights are obtained by solving a singlelinear system. For $\mathbb{S}^2$, and the restricted thin plate spline kernel$r^2 \log r$, these weights can be computed for two-thirds of a million nodes,using a preconditioned iterative technique introduced by us.

Kernel Based Quadrature on Spheres and Other Homogeneous Spaces