Approximation of discrete functions and size of spectrum

Abstract

Let $\Lambda \subset \mathbb{R}$ be a uniformly discrete sequence and $S \subset \mathbb{R}$ a compact set. We prove that if there exists a bounded sequence of functions in Paley-Wiener space $PW_s$, which approximates $\delta$-functions on $\Lambda$ with $l^2$-error $d$, then measure$(S) \geq 2\pi(1-d^2)D^+(\Lambda)$. This estimate is sharp for every $d$. Analogous estimate holds when the norms of approximating functions have a moderate growth, and we find a sharp growth restriction.