Harmonic analysis invariants for infinite graphs via operators and algorithms

TL;DR

This paper explores harmonic analysis invariants on infinite graphs using operator theory and algorithms, linking graph theory with potential theory, Fourier analysis, and boundary concepts to extend classical sampling theories.

Contribution

It introduces new methods combining combinatorial and operator-theoretic tools to analyze infinite graphs and their harmonic functions, extending Shannon's sampling theory to these structures.

Findings

Development of spectral invariants for infinite graphs

Connection between graph analysis and classical potential theory

Extension of sampling/interpolation concepts to infinite graph settings

Abstract

We present recent advances in harmonic analysis on infinite graphs. Our approach combines combinatorial tools with new results from the theory of unbounded Hermitian operators in Hilbert space, geometry, boundary constructions, and spectral invariants. We focus on particular classes of infinite graphs, including such weighted graphs which arise in electrical network models, as well as new diagrammatic graph representations. We further stress some direct parallels between our present analysis on infinite graphs, on the one hand, and, on the other, specific areas of potential theory, Fourier duality, probability, harmonic functions, sampling/interpolation, and boundary theory. With the use of limit constructions, finite to infinite, and local to global, we outline how our results for infinite graphs may be viewed as extensions of Shannon's theory: Starting with a countable infinite graph $G$, and a suitable fixed positive weight function, we show that there are certain continua (certain ambient sets $X$) extending $G$, and associated notions of interpolation for (Hilbert spaces of) functions on $X$ from their restrictions to the discrete graph $G$.