Our article is a summary of some results for Riemannian manifolds that were obtained in \cite{ gpes}-\cite{ Pesssubm}. To the best of our knowledge these are the pioneering papers which contain the most general results about frames, Shannon sampling, and cubature formulas on compact and non-compact Riemannian manifolds. In particular, the paper \cite{ gpes} gives an "end point" construction of tight localized frames on homogeneous compact manifolds. The paper \cite{ Pessubm} is the first systematic development of localized frames on compact domains in Euclidean spaces.
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