Positive definite kernels and boundary spaces

TL;DR

This paper develops a generalized harmonic analysis framework using positive definite kernels to define measure-theoretic boundaries, extending classical results to discrete and sampling-based settings.

Contribution

It introduces a measure-theoretic boundary concept for positive definite kernels and proves their existence using Gaussian process theory, broadening harmonic analysis applications.

Findings

Existence of measure-theoretic boundaries for positive definite kernels

Extension of classical harmonic analysis to discrete and sampling contexts

Framework applicable to electrical networks and sampling operations

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.