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

Shannon Sampling and Parseval Frames on Compact Manifolds