The main goal of the article is to show that Paley-Wiener functions ƒ ∈ L2(M) of a fixed band width to on a Riemannian manifold of bounded geometry M completely determined and can be reconstructed from a set of numbers Φi (ƒ), i ∈ ℕwhere Φiis a countable sequence of weighted integrals over a collection of “small” and “densely” distributed compact subsets. In particular, Φi, i ∈ ℕ,can be a sequence of weighted Dirac measures δxi, xi ∈M.
It is shown that Paley-Wiener functions on M can be reconstructed as uniform limits of certain variational average spline functions.
To obtain these results we establish certain inequalities which are generalizations of the Poincaré-Wirtingen and Plancherel-Polya inequalities.
Our approach to the problem and most of our results are new even in the one-dimensional case.
Math Subject Classifications
Key Words and Phrases
Poincaré inequality on manifoldsLaplace-Beltrami operatorband-limited functions on manifoldssplines on manifoldssampling theorem on manifolds