An extension of Mercer theorem to vector-valued measurable kernels

TL;DR

This paper generalizes Mercer’s theorem to vector-valued kernels in measurable spaces, providing a series representation of the kernel using eigenfunctions under mild conditions, with convergence properties on compact sets.

Contribution

It extends Mercer’s theorem to vector-valued measurable kernels, establishing eigenfunction expansions and convergence results in this broader setting.

Findings

Kernel represented as a convergent series of eigenfunctions

Eigenfunctions are continuous with respect to a natural topology

Series converges uniformly on compact subsets when support of measure is the entire space

Abstract

We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space $X$into $\mathbb C^n$. Given a finite measure $\mu$ on $X$, we represent the reproducing kernel $K$ as convergent series in terms of the eigenfunctions of a suitable compact operator depending on $K$ and $\mu$. Our result holds under the mild assumption that $K$ is measurable and the associated Hilbert space is separable. Furthermore, we show that $X$ has a natural second countable topology with respect to which the eigenfunctions are continuous and the series representing $K$ uniformly converges to $K$ on any compact subsets of $X\times X$, provided that the support of $\mu$ is $X$.