Sampling in paley-wiener spaces on combinatorial graphs

TL;DR

This paper introduces Paley-Wiener spaces on combinatorial graphs, characterizes uniqueness sets, and develops a reconstruction algorithm using frame theory, with applications to lattices, trees, and finite graphs.

Contribution

It extends Paley-Wiener space theory to combinatorial graphs and proposes a novel reconstruction method based on frames and Poincare-Wirtinger inequalities.

Findings

Uniqueness sets are characterized by Poincare-Wirtinger inequalities.

A frame-based reconstruction algorithm is developed.

Applications to lattices, trees, and finite graphs are demonstrated.

Abstract

A notion of Paley-Wiener spaces is introduced on combinatorial graphs. It is shown that functions from some of these spaces are uniquely determined by their values on some sets of vertices which are called the uniqueness sets. Such uniqueness sets are described in terms of Poincare-Wirtingertype inequalities. A reconstruction algorithm of Paley-Wiener functions from uniqueness sets which uses the idea of frames in Hilbert spaces is developed. Special consideration is given to n-dimensional lattice, homogeneous trees, and eigenvalue and eigenfunction problems on finite graphs.