Reproducing Kernel Hilbert Spaces and entropy Kolmogorov numbers on compact Lie Groups

TL;DR

This paper investigates the asymptotic behavior of entropy Kolmogorov numbers for embeddings of reproducing kernel Hilbert spaces on compact Lie groups, providing bounds that enhance understanding of function space complexity.

Contribution

It offers new asymptotic estimates for entropy Kolmogorov numbers of RKHS embeddings on compact Lie groups, linking harmonic analysis with approximation theory.

Findings

Derived lower and upper bounds for entropy Kolmogorov numbers

Established asymptotic estimates for embeddings of RKHS into continuous functions

Connected spectral properties of integral operators with function space complexity

Abstract

On a compact Lie group $G$, we consider the reproducing kernel Hilbert space $\mathcal{H}_K$ associated with the integral kernel $K$ of a left-invariant, positive, symmetric, trace class integral operator on $L^2(G)$. We present lower and upper asymptotic estimates for the entropy Kolmogorov numbers (also called covering numbers) for the embedding of $\mathcal{H}_K$ into the space $C(G)$ of continuous functions on $G$.