  We consider the problem of positive-semidefinite continuation: extending a
partially specified covariance kernel from a subdomain $\Omega$ of a
rectangular domain $I\times I$ to a covariance kernel on the entire domain
$I\times I$. For a broad class of domains $\Omega$ called \emph{serrated
domains}, we are able to present a complete theory. Namely, we demonstrate that
a canonical completion always exists and can be explicitly constructed. We
characterise all possible completions as suitable perturbations of the
canonical completion, and determine necessary and sufficient conditions for a
unique completion to exist. We interpret the canonical completion via the
graphical model structure it induces on the associated Gaussian process.
Furthermore, we show how the estimation of the canonical completion reduces to
the solution of a system of linear statistical inverse problems in the space of
Hilbert-Schmidt operators, and derive rates of convergence. We conclude by
providing extensions of our theory to more general forms of domains, and by
demonstrating how our results can be used to construct covariance estimators
from sample path fragments of the associated stochastic process. Our results
are illustrated numerically by way of a simulation study and a real example.
