Widths and entropy of sets of smooth functions on compact homogeneous manifolds

Abstract

We develop a general method to calculate entropy and n-widths of sets of smooth functions on an arbitrary compact homogeneous Riemannian manifold M-d. Our method is essentially based on a detailed study of geometric characteristics of norms induced by subspaces of harmonics on M-d. This approach has been developed in the cycle of works [1, 2, 10-19]. The method's possibilities are not confined to the statements proved but can be applied in studying more general problems. As an application, we establish sharp orders of entropy and n-widths of Sobolev's classes W-p(gamma)(M-d) and their generalisations in L-q (M-d) for any 1 < p, q < infinity. In the case p, q = 1, infinity sharp in the power scale estimates are presented.