Super-exponential decay rates for eigenvalues and singular values of integral operators on the sphere

TL;DR

This paper establishes super-exponential decay rates for eigenvalues and singular values of integral operators on the sphere with smooth kernels, providing theoretical insights and practical kernel examples for numerical and geostatistical applications.

Contribution

It introduces super-exponential decay results for eigenvalues and singular values of sphere-based integral operators under Laplace-Beltrami smoothness assumptions, including optimality and kernel examples.

Findings

Super-exponential decay rates for eigenvalues and singular values.

Identification of kernel families satisfying smoothness assumptions.

Application relevance to numerical analysis and geostatistics.

Abstract

This paper brings results about the behavior of sequences of eigenvalues or singular values of integral operators generated by square-integrable kernels on the real m-dimensional unit sphere, $m\leq2$. Under smoothness assumptions on the generating kernels, given via Laplace-Beltrami differentiability, we obtain super-exponential decay rates for the eigenvalues of the generated positive integral operators and for singular values of those integral operators which are non-positive. We show an optimal-type result and provide a list of parametric families of kernels which are of interest for numerical analysis and geostatistical communities and satisfy the smoothness assumptions for the positive case.