The Mercer-Young Theorem for Matrix-Valued Kernels on Separable Metric Spaces

TL;DR

This work extends the classical Mercer-Young theorem to matrix-valued kernels on separable metric spaces, providing a complete equivalence between three positivity notions for $K:X×X→R^{N×N}$ with symmetry $K(x,y)=K(y,x)^T$ under a locally finite measure. The authors prove (ii)⇒(i) on general domains by adapting Mercer's geometric argument with Urysohn's lemma, and (i)⇒(iii) using a matrix-valued Mercer decomposition (Devito 2013), first for finite measures and then for locally finite ones. The resulting framework unifies scalar and matrix-valued kernel theory and has implications for potential theory, convex optimization, and kernel-based discretizations in PDEs, including multi-task and numerical applications. By closing a gap for general domains and codomains, the paper strengthens the theoretical foundation for using matrix-valued kernels in broad applied settings and provides a pathway for efficient finite-dimensional approximations via integral positivity controls.

Abstract

We generalize the characterization theorem going back to Mercer and Young, which states that a symmetric and continuous kernel is positive definite if and only if it is integrally positive definite, to matrix-valued kernels on separable metric spaces. We also demonstrate the applications of the generalized theorem to the field of convex optimization and other areas.

Figures (2)

• Figure 1: Functions $f_{x_i,\delta,\varepsilon}$ corresponding to given points $x_i\in X$ for $i\in\{1,\dots,n\}$ and given $\delta,\varepsilon>0$ in the case where $X=[a,b]$ and $n=3$.
• Figure 2: The subsets $Q_{ij}$, $q_{ij}$, $r_{ij}\subset X\times X$ associated with given points $x_i,x_j\in X$ for $i,j\in\{1,\dots,n\}$ in the case where $X=[a,b]$ and $n=3$.

Theorems & Definitions (6)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Lemma 2.1
• proof : Proof of Lemma \ref{['lemma:equivalence of real and complex']}
• proof : Proof of Theorem \ref{['thm:generalization']}