Entropy of Sobolev's classes on Compact Homogeneous Riemannian Manifolds

TL;DR

This paper introduces a general method for calculating entropy numbers of Sobolev classes on compact homogeneous Riemannian manifolds, leveraging geometric properties of harmonic subspaces, with applications to entropy and n-widths.

Contribution

A novel, broadly applicable method for entropy calculation on manifolds, extending to multiplier operators and providing sharp entropy and n-width estimates.

Findings

Established sharp entropy orders for Sobolev classes

Derived precise n-widths for these classes

Demonstrated method's applicability to multiplier operators

Abstract

We develop a general method to calculate entropy numbers of standard Sobolev's classes on an arbitrary compact homogeneous Riemannian manifold. Our method is essentially based on a detailed study of geometric characteristics of norms induced by subspaces of harmonics. The method's possibilities are not confined to the statements proved but can be applied in studying more general problems such as entropy of multiplier operators. As an application, we establish sharp orders of entropy of Sobolev's classes and respective n-widths.