Shannon Sampling and Parseval Frames on Compact Manifolds

The problem of representation and analysis of functions defined on manifolds (signals, images, and data in general) is ubiquities in many fields ranging from statistics and cosmology to neuroscience and biology. It is very common to consider input signals as points in a high-dimensional measurement space, however, meaningful structures lay on a manifold embedded in this space. In the last decades, the importance of these applications triggered the development of various generalized wavelet bases suitable for the unit spheres S and S and the rotation group of R. The goal of the present study is to describe a general approach to bandlimited localized Parseval frames in a space L2(M), where M is a compact homogeneous Riemannian manifold. One can think of a Riemannian manifold as of a surface in a Euclidean space. A homogeneous manifold is a surface with ”many” symmetries like the sphere x1 + ...+ x 2 d = 1 in Euclidean space R. Our construction of frames in a function space L2(M) heavily depends on proper notions of bandlimitedness and Shannon-type sampling on a manifold M. The crucial role in this development is played by positive cubature formulas (Theorem 1.3) and by the product property (Theorem 1.2), which were proved in [1] and [10]. The notion of bandlimideness on a compact manifold M is introduced in terms of eigenfunctions of a certain secondorder differential elliptic operator on M. The most important fact for our construction of frames is that in a space of ω-bandlimited functions the regular L2(M) norm can be descretized. This result in the case of compact manifolds (and even non-compact manifolds of bounded geometry) was first discovered and explored in many ways in our papers [?]-[?]. In the classical cases of straight line R and circle S the corresponding results are known as Plancherel-Polya and Marcinkiewicz-Zygmund inequalities. Our generalization of Plancherel-Polya and Marcinkiewicz-Zygmund inequalities implies that ω-bandlimited functions on manifolds are completely determined by their vales on discrete sets of points ”uniformly” distributed over M with a spacing comparable to 1/ √ ω and can be completely reconstructed in a stable way from their values on such sets. The last statement is an extension of the Shannon sampling theorem to the case of Riemannian manifolds. Our article is a summary of some results for Riemannian manifolds that were obtained in [1]-[12]. 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 [1] gives an ”end point” construction of tight localized frames on homogeneous compact manifolds. The paper [11] is the first systematic development of localized frames on compact domains in Euclidean spaces.