Abstract: Given a Riemannian manifold X with Riemannian measure ?X and positive weights {?j}j=1N, we study the conditions under which there exist points {xj}j=1N?X so that a cubature formula of the form(1)?XPd?X=?j=1N?jP(xj)holds for all polynomials P of order less than or equal to L. The problem is studied for diffusion polynomials (linear combinations of eigenfunctions of the Laplace-Beltrami operator) in the context of abstract Riemannian manifolds and for algebraic polynomials in the context of algebraic manifolds in Rn.

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