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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Approximation of discrete functions and size of spectrum

Authors: A. Olevskiĭ and A. Ulanovskiĭ
Original publication: Algebra i Analiz, tom 21 (2009), nomer 6.
Journal: St. Petersburg Math. J. 21 (2010), 1015-1025
MSC (2010): Primary 30D15, 42A16
Published electronically: September 22, 2010
MathSciNet review: 2604548
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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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Additional Information

A. Olevskiĭ
Affiliation: School of Mathematics, Tel Aviv University, Ramat Aviv 69978, Israel

A. Ulanovskiĭ
Affiliation: Stavanger University, Stavanger 4036, Norway

Keywords: Paley–Wiener space, Bernstein space, set of interpolation, approximation of discrete functions
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.
Article copyright: © Copyright 2010 American Mathematical Society

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