  We consider a kernel based harmonic analysis of "boundary," and boundary
representations. Our setting is general: certain classes of positive definite
kernels. Our theorems extend (and are motivated by) results and notions from
classical harmonic analysis on the disk. Our positive definite kernels include
those defined on infinite discrete sets, for example sets of vertices in
electrical networks, or discrete sets which arise from sampling operations
performed on positive definite kernels in a continuous setting.
  Below we give a summary of main conclusions in the paper: Starting with a
given positive definite kernel $K$ we make precise generalized boundaries for
$K$. They are measure theoretic "boundaries." Using the theory of Gaussian
processes, we show that there is always such a generalized boundary for any
positive definite kernel.
