Approximation of discrete functions and size of spectrum

Let $\Lambda\subset\mathbb R$ be a uniformly discrete sequence and $S\subset\mathbb R$ a compact set. It is proved that if there exists a bounded sequence of functions in the Paley–Wiener space $PW_S$ that approximates $\delta$-functions on $\Lambda$ with $l^2$-error $d$, then the measure of $S$ cannot be less than $2\pi(1-d^2)D^+(\Lambda)$. This estimate is sharp for every $d$. A similar estimate holds true when the norms of approximating functions have a moderate growth; the corresponding sharp growth restriction is found.