  Gaussian processes are among the most useful tools in modeling continuous
processes in machine learning and statistics. If the value of a process is
known at a finite collection of points, one may use Gaussian processes to
construct a surface which interpolates these values to be used for prediction
and uncertainty quantification in other locations. However, it is not always
the case that the available information is in the form of a finite collection
of points. For example, boundary value problems contain information on the
boundary of a domain, which is an uncountable collection of points that cannot
be incorporated into typical Gaussian process techniques. In this paper we
construct a Gaussian process model which utilizes reproducing kernel Hilbert
spaces to unify the typical finite case with the case of having uncountable
information by exploiting the equivalence of conditional expectation and
orthogonal projections. We discuss this construction in statistical models,
including numerical considerations and a proof of concept.
