Conference Agenda

Mercer's celebrated theorem is refined and extended for (weakly) differentiable symmetric kernels by associating not the common $L^2$-integral operator but a slightly more complex operator, that additionally takes into account information encoded in the (weak) derivatives of the kernel. The natural domain for this associated operator is the Sobolev Space $H^k(\Theta) = W^{k,2}(\Theta) \subset L^2(\Theta)$, where $\Theta \subset \R^d$ is some bounded domain and $k\in\N_0$ depends on the order of weak differentiability. The spectral decomposition of this operator then leads to a Mercer-type expansion of the kernel, which converges with respect to the $H^k$-norm and, if $k>d$, also uniformly \emph{without} requiring the kernel to be positive-definite. In case the kernel is also positive-definite and differentiable in the strong sense, a refinement of Mercer's theorem is obtained that additionally provides uniform convergence of the term-wise derivatives of the expansion to the respective derivatives of the kernel as well.\\
Finally, applied to the covariance kernel of a (weakly) differentiable stochastic process, these results provide a novel Karhunen-Loève-type representation that simultaneously approximates the process as well as its (weak) derivatives in a mean square optimal sense.