# Isotropic covariance matrix functions on compact two-point homogeneous spaces

## TL;DR

This paper characterizes isotropic covariance matrix functions on compact two-point homogeneous spaces, providing necessary and sufficient conditions and linking Euclidean and non-Euclidean cases.

## Contribution

It introduces a comprehensive characterization of isotropic covariance functions on these spaces, extending understanding beyond Euclidean settings.

## Key findings

- Necessary and sufficient conditions for covariance functions
- Equivalence of covariance functions on Euclidean and spherical spaces
- Extension to compact two-point homogeneous spaces

## Abstract

The covariance matrix function is characterized in this paper for a Gaussian or elliptically contoured vector random field that is stationary, isotropic, and mean square continuous on the compact two-point homogeneous space. Necessary and sufficient conditions are derived for a symmetric and continuous matrix function to be an isotropic covariance matrix function on all compact two-point homogeneous spaces. It is also shown that, for a symmetric and continuous matrix function with compact support, if it makes an isotropic covariance matrix function in the Euclidean space, then it makes an isotropic covariance matrix function on the sphere or the real projective space.