Positive definite kernels and boundary spaces

Document Type: Original Article



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.