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Trigonometric polynomials with many real zeros and a Littlewood-type problem

Authors: Peter Borwein and Tamás Erdélyi
Journal: Proc. Amer. Math. Soc. 129 (2001), 725-730
MSC (2000): Primary 41A17
Published electronically: November 3, 2000
MathSciNet review: 1801998
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Abstract | References | Similar Articles | Additional Information

Abstract: We examine the size of a real trigonometric polynomial of degree at most $n$ having at least $k$ zeros in $K := {\mathbb{R}} (\text{mod} 2\pi )$ (counting multiplicities). This result is then used to give a new proof of a theorem of Littlewood concerning flatness of unimodular trigonometric polynomials. Our proof is shorter and simpler than Littlewood's. Moreover our constant is explicit in contrast to Littlewood's approach, which is indirect.

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Additional Information

Peter Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

Tamás Erdélyi
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Keywords: Real trigonometric polynomials, real zeros, unimodular trigonometric polynomials, flatness
Received by editor(s): March 2, 1999
Published electronically: November 3, 2000
Additional Notes: The research of the first author was supported, in part, by NSERC of Canada. The research of the second author was supported, in part, by the NSF under Grant No. DMS–9623156.
Communicated by: David R. Larson
Article copyright: © Copyright 2000 Copyright retained by the authors