Kolmogorov widths on the sphere via eigenvalue estimates for H\"{o}lderian integral operators

TL;DR

This paper establishes sharp upper bounds for Kolmogorov widths on the sphere by analyzing eigenvalue decay rates of integral operators associated with H"olderian kernels, advancing approximation theory in RKHS.

Contribution

It provides a novel approach to estimate Kolmogorov widths from eigenvalue decay, specifically for kernels satisfying an abstract H"older condition, with sharp upper bounds.

Findings

Derived sharp upper bounds for Kolmogorov widths.

Connected eigenvalue decay rates to approximation quality.

Applied to kernels on the sphere with H"older continuity.

Abstract

Approximation processes in the reproducing kernel Hilbert space associated to a continuous kernel on the unit sphere $S^m$ in the Euclidean space $\mathbb{R}^{m+1}$ are known to depend upon the Mercer's expansion of the compact and self-adjoint $L^2(S^m)$-operator associated to the kernel. The estimation of the Kolmogorov $n$-th width of the unit ball of the reproducing kernel Hilbert space in $L^2(S^m)$ and the identification of the so-called optimal subspace usually suffice. These Kolmogorov widths can be computed through the eigenvalues of the integral operator associated to the kernel. This paper provides sharp upper bounds for the Kolmogorov widths in the case in which the kernel satisfies an abstract H\"{o}lder condition. In particular, we follow the opposite direction usually considered in the literature, that is, we estimate the widths from decay rates for the sequence of eigenvalues of the integral operator.