Positive definite kernels and boundary spaces

Abstract

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.