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\rcompact homogeneous Riemannian manifold Md\r. Our method is essentially based on a detailed study of geometric\rcharacteristics of norms induced by subspaces of harmonics on Md\r. This approach has been developed in the cycle\rof works [1, 2, 10–19]. The method’s possibilities are not confined to the statements proved but can be applied in\rstudying more general problems. As an application, we establish sharp orders of entropy and n-widths of Sobolev’s\rclasses Wγ\rp\r(\rMd\r)\rand their generalisations in Lq\r(\rMd\r)\rfor any 1 < p, q < ∞. In the case p, q = 1, ∞ sharp in the power\rscale estimates are presented.