Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Positive definite singular kernels on two-point homogeneous spaces

Dmitriy Bilyk, Peter Grabner

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.

Positive definite singular kernels on two-point homogeneous spaces

Abstract

We study positive definiteness of kernels 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 . Specifically, the Riesz kernel (where 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.

Paper Structure

This paper contains 6 sections, 12 theorems, 73 equations, 1 table.

Table of Contents

  1. Introduction
  2. Harmonic analysis on two-point homogeneous spaces
  3. Positive definite functions
  4. Schur's lemma for singular integrable kernels
  5. Functions that are positive definite for all $\mathbb{FP}^{d-1}$
  6. Riesz kernels for the geodesic and chordal distance

Key Result

Theorem 1

Let $F:[-1,1)\to\mathbb{R}$ be a function in $L^1(\mu_{\alpha,\beta}) \cap \mathop{\mathrm{\mathrm{C}}}\nolimits([-1,1))$. Then $F$ is positive definite if and only if all of its Fourier coefficients are non-negative.

Theorems & Definitions (28)

  • Definition 1
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof : Proof of Theorem \ref{['thm:energyminimize']}
  • Theorem 5
  • proof
  • ...and 18 more