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.References
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Bibliographic Information
- A. Olevskiĭ
- Affiliation: School of Mathematics, Tel Aviv University, Ramat Aviv 69978, Israel
- MR Author ID: 224313
- Email: olevskii@post.tau.ac.il
- A. Ulanovskiĭ
- Affiliation: Stavanger University, Stavanger 4036, Norway
- MR Author ID: 194862
- Email: Alexander.Ulanovskii@uis.no
- Received by editor(s): August 20, 2009
- Published electronically: September 22, 2010
- Additional Notes: The first author was partially supported by the Israel Science Foundation.
- © Copyright 2010 American Mathematical Society
- Journal: St. Petersburg Math. J. 21 (2010), 1015-1025
- MSC (2010): Primary 30D15, 42A16
- DOI: https://doi.org/10.1090/S1061-0022-2010-01129-4
- MathSciNet review: 2604548