Approximation of discrete functions and size of spectrum

Abstract

Let $\Lambda$ be a uniformly discrete set and $S$ be a compact set in $R$. 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.