Approximation of discrete functions and size of spectrum

Olevskiĭ, A.; Ulanovskiĭ, A.

doi:10.1090/S1061-0022-2010-01129-4

Approximation of discrete functions and size of spectrum
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by A. Olevskiĭ and A. Ulanovskiĭ

St. Petersburg Math. J. 21 (2010), 1015-1025

DOI: https://doi.org/10.1090/S1061-0022-2010-01129-4

Published electronically: September 22, 2010

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Abstract:

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 the approximating functions have a moderate growth; the corresponding sharp growth restriction is found.

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