Graph Laplacians and discrete reproducing kernel Hilbert spaces from restrictions

TL;DR

This paper explores the relationship between kernel functions, reproducing kernel Hilbert spaces, and graph Laplacians on infinite discrete sets, using continuous models as analytical tools for discrete problems.

Contribution

It introduces a novel approach that analyzes infinite discrete models via continuous counterparts to derive solutions and insights.

Findings

Establishes connections between graph Laplacians and RKHS on infinite sets

Develops methods for analyzing discrete models through continuous analogs

Provides a framework for solving boundary value problems on discrete structures

Abstract

We study kernel functions, and associated reproducing kernel Hilbert spaces $\mathscr{H}$ over infinite, discrete and countable sets $V$. Numerical analysis builds discrete models (e.g., finite element) for the purpose of finding approximate solutions to boundary value problems; using multiresolution-subdivision schemes in continuous domains. In this paper, we turn the tables: our object of study is realistic infinite discrete models in their own right; and we then use an analysis of suitable continuous counterpart problems, but now serving as a tool for obtaining solutions in the discrete world.