Sampling on Paley-Wiener spaces on graphs, with particular focus on the infinite-dimensional case

TL;DR

This paper establishes a sampling theorem for infinite-dimensional Paley-Wiener spaces on graphs, enabling stable reconstruction and characterizing sampling sets as complements of lambda-sets.

Contribution

It introduces a new sampling theorem for infinite-dimensional graph-based Paley-Wiener spaces and characterizes sampling sets via lambda-sets for stable reconstruction.

Findings

Sampling sets are complements of lambda-sets.

Provides sufficient conditions for stable sampling on various graphs.

Extends classical sampling theory to infinite-dimensional graph spaces.

Abstract

We prove a sampling theorem for infinite-dimensional Paley-Wiener spaces on graphs which allows for stable frame reconstruction. We prove that all sampling sets for a fixed Paley-Wiener space are complements of lambda-sets (i.e. sets where a Poincar\'e-type inequality holds), thereby providing a sufficient condition for stable sampling and reconstruction on graphs such as $\mathbb{Z}^n$-lattices and radial trees with finite geometry.