Jackson kernels: a tool for analyzing the decay of eigenvalue sequences of integral operators on the sphere

TL;DR

This paper introduces Jackson kernels as a new analytical tool to estimate the decay rates of eigenvalues for positive integral operators on the sphere, extending classical results to a spherical setting.

Contribution

It presents a novel approach using generalized Jackson kernels to analyze eigenvalue decay rates for integral operators on the sphere, broadening existing theoretical frameworks.

Findings

Eigenvalue decay rates match classical H"older conditions

The approach extends classical results to spherical integral operators

Provides estimations based on properties of generalized Jackson kernels

Abstract

Decay rates for the sequence of eigenvalues of positive and compact integral operators has been largely investigated for a long time in the literature. In this paper, the focus will be on positive integral operators acting on square integrable functions on the unit sphere and generated by a kernel satisfying a H\"older type assumption defined via average operators. In the approach to be presented here, the decay rates will be reached from convenient estimations on the eigenvalues of the operator themselves, with the help of specific properties of a generic approximation operator defined through the so-called generalized Jackson kernels. The decay rates have the same structure of those known to hold in the cases in which the H\"older condition is the classical one. Therefore, within the spherical setting, the abstract approach to be introduced here extends some classical results on the topic.