Positive definite singular kernels on two-point homogeneous spaces

TL;DR

This paper investigates positive definite kernels with singularities on two-point homogeneous spaces, extending classical results to include kernels like Riesz kernels with singularities, and characterizes their positive definiteness.

Contribution

It extends classical positive definiteness results to kernels with singularities on two-point homogeneous spaces, including Riesz kernels, and provides new characterizations.

Findings

Derived analogs of Schoenberg's and Schur's lemmas for singular kernels.

Characterized positive definiteness of Riesz kernels on projective spaces.

Enhanced understanding of kernels with singularities in geometric analysis.

Abstract

We study positive definiteness of kernels $K(x,y)$ on two-point homogeneous spaces. As opposed to the classical case, which has been developed and studied in the existing literature, we allow the kernel to have an (integrable) singularity for $x=y$. Specifically, the Riesz kernel $d(x,y)^{-s}$ (where $d$ denotes some distance on the space) is a prominent example. We derive results analogous to Schoenberg's characterization of positive definite functions on the sphere, Schur's lemma on the positive definiteness of the product of positive definite functions, and Schoenberg's characterization of functions positive definite on all spheres. We use these results to better understand the behavior of the Riesz kernels for the geodesic and chordal distances on projective spaces.