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

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.