Sobolev-Mercer Expansions and Applications to Stochastic Processes

We establish a fundamental extension of Mercer's celebrated theorem by introducing a class of higher-order kernel operators acting on Sobolev spaces $H^k(\Theta)$, where $\Theta \subset \mathbb{R}^d$ is a bounded domain and $k\in\mathbb{N}_0$ corresponds to the order of weak differentiability. The spectral decomposition of these operators then yields Mercer-type expansions that are optimal in $H^k(\Theta\times\Theta)$. Notably, we derive from the embedding properties of Sobolev spaces, that for $k>d$, these expansions also converge uniformly without requiring the kernel to be positive definite. For positive definite kernels, we confirm the nuclearity of these higher-order operators and establish a significant refinement of Mercer's Theorem. These results lead to novel spectral representations of RKHS and have subtle implications for stochastic analysis. Applied to the covariance kernels of weakly differentiable random fields, our theory provides refined Karhunen-Loeve expansions that facilitate the simultaneous mean-square optimal approximation of both the process and its derivatives.