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

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 ≥ 2. Under smoothness assumptions on the kernels, given via (infinitely many times) Laplace-Beltrami differentiability, we obtain super-exponential decay rates for the eigenvalues of positive integral operators and for singular values of those which are non-positive. The positive case is shown to be optimal. We also provide a list of parametric families of these kernels, which are of interest for numerical analysis and geostatistical communities.MSC: 45C05 (41A36 42A82 45M05 45P05 47A75)