On approximation tools and its applications on compact homogeneous spaces

TL;DR

This paper characterizes approximation tools on compact homogeneous spaces, extending spherical results, and applies these to eigenvalue decay and Kolmogorov n-width estimates in RKHS.

Contribution

It extends Peetre type K-functional characterization to compact homogeneous spaces and links kernel smoothness to eigenvalue decay and approximation rates.

Findings

Characterization of K-functional on compact homogeneous spaces.

Eigenvalue decay rates for kernels with smoothness conditions.

Estimates for Kolmogorov n-width in RKHS.

Abstract

We prove a characterization for the Peetre type $K$-functional on $\mathbb{M}$, a compact two-point homogeneous space, in terms the rate of approximation of a family of multipliers operator defined to this purpose. This extends the well known results on the spherical setting. The characterization is employed to show that an abstract H\"{o}lder condition or finite order of differentiability condition imposed on kernels generating certain operators implies a sharp decay rates for their eigenvalues sequences. The latest is employed to obtain estimates for the Kolmogorov $n$-width of unit balls in Reproducing Kernel Hilbert Space (RKHS).