Approximation of discrete functions and size of spectrum

Abstract

Let Λ⊂R be a uniformly discrete sequence and S⊂R a compact set. We prove that if there exists a bounded sequence of functions in Paley-Wiener space PWs, which approximates δ-functions on Λ with l2-error d, then measure(S)≥2π(1−d2)D+(Λ). 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.