Function recovery on manifolds using scattered data

We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold M when given a sample on a finite point set. We prove that the quality of the sample is given by the L_ (M) -average of the geodesic distance to the point set and determine the value of (0, ]. This extends our findings on bounded convex domains arXiv: 2009. 11275, 2020. Further, a limit theorem for moments of the average distance to a set consisting of i. i. d. uniform points is proven. This yields that a random sample is asymptotically as good as an optimal sample in precisely those cases with <. In particular, we obtain that cubature formulas with random nodes are asymptotically as good as optimal cubature formulas if the weights are chosen correctly. This closes a logarithmic gap left open by Ehler, Graf and Oates Stat. Comput. , 29: 1203-1214, 2019.