Estimates of Kolmogorov, Gelfand and linear $n$- widths on Compact Riemannian Manifolds

TL;DR

This paper provides precise estimates for Kolmogorov, Gelfand, and linear n-widths of Sobolev unit balls in Lp spaces on compact Riemannian manifolds, extending previous results to general cases.

Contribution

It establishes lower and exact asymptotic estimates for n-widths on compact Riemannian manifolds, generalizing prior homogeneous manifold results.

Findings

Lower and exact estimates of n-widths are obtained.

Estimates are asymptotically exact for homogeneous manifolds.

Proofs use kernel localization of elliptic operators.

Abstract

We determine lower and exact estimates of Kolmogorov, Gelfand and linear $n$-widths of unit balls in Sobolev norms in $L_{p}$-spaces on compact Riemannian manifolds. As it was shown by us previously these lower estimates are exact asymptotically in the case of compact homogeneous manifolds. The proofs rely on two-sides estimates for the near-diagonal localization of kernels of functions of elliptic operators.