Eigenvalue decay of positive integral operators on compact two-point homogeneous spaces

TL;DR

This paper extends the understanding of eigenvalue decay rates for positive integral operators from spheres to more general two-point homogeneous spaces, linking decay rates to geometric and operator properties.

Contribution

It generalizes existing results on eigenvalue decay to a broader class of spaces and relates decay rates to the Laplace-Beltrami operator and Schatten class conditions.

Findings

Eigenvalue decay rates depend on the Laplace-Beltrami operator order.

Decay rates are influenced by the Schatten class of the operator.

Results apply to a wide class of two-point homogeneous spaces.

Abstract

We generalize and extend results on decay rates of singular values or eigenvalues of positive integral operators from unit spheres to two-point homogeneous spaces. The rates we present depend upon the order of the Laplace-Beltrami operator used to define the smoothness conditions on generating kernels, the Schatten class containing the integral operator generated by the derivative of the generating kernel and the dimension of the space.