Overview
This work proposes an expansion of Mercer's theorem to a class of higher-order kernel operators. These operators function on Sobolev spaces, denoted as $H^k(\Theta)$, where $\Theta \subset \mathbb{R}^d$ represents a bounded domain, and $k\in\mathbb{N}_0$ signifies the order of weak differentiability.
Research Context
Mercer's theorem traditionally provides a spectral decomposition for continuous, symmetric, positive-definite kernels over compact domains. This decomposition is fundamental in various areas, including the construction of Reproducing Kernel Hilbert Spaces (RKHS) and the spectral analysis of stochastic processes. The current research aims to extend this foundational theorem to higher-order settings, specifically within the framework of Sobolev spaces.
Approach
The study establishes a fundamental extension of Mercer's theorem by introducing a class of higher-order kernel operators. The spectral decomposition of these operators is then analyzed to derive Mercer-type expansions. The methodology involves examining the embedding properties of Sobolev spaces to determine convergence characteristics of these expansions. For positive definite kernels, the research confirms the nuclearity of these higher-order operators.
Findings
- The developed Mercer-type expansions are optimal in $H^k(\Theta\times\Theta)$.
- From the embedding properties of Sobolev spaces, it was derived that for $k>d$, these expansions converge uniformly. This uniform convergence occurs even without the requirement for the kernel to be positive definite.
- For positive definite kernels, the nuclearity of the higher-order operators was confirmed.
- A refinement of Mercer's Theorem was established for positive definite kernels in this higher-order context.
- These results contribute to novel spectral representations of RKHS.
- The theory has implications for stochastic analysis.
- Applied to the covariance kernels of weakly differentiable random fields, the theory yields refined Karhunen-Loeve expansions. These expansions facilitate the simultaneous mean-square optimal approximation of both the stochastic process and its derivatives.
Why This Matters
The theoretical advancements provide a refined framework for understanding and representing functions within Sobolev spaces through spectral decompositions. The derived Karhunen-Loeve expansions offer a method for simultaneously approximating stochastic processes and their derivatives in a mean-square optimal sense, which can be relevant in fields involving stochastic analysis and modeling.
Potential Applications
The theory provides refined Karhunen-Loeve expansions for the covariance kernels of weakly differentiable random fields. This can facilitate the simultaneous mean-square optimal approximation of a stochastic process and its derivatives.