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Approximation of discrete functions and size of spectrum
2009, Olevskii, Alexander, Ulanovskii, Alexander
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.