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arXiv · 1304.0649

Approximation of discrete functions and size of spectrum

Alexander Olevskii·Alexander Ulanovskii
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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.

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Alexander Olevskii, Alexander Ulanovskii. 2013-04-02. Approximation of discrete functions and size of spectrum. https://arxiv.org/abs/1304.0649

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