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

DOI:
https://doi.org/10.1090/S0002-9939-00-06021-4

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 having at least zeros in (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

Email:
pborwein@cecm.sfu.ca

**Tamás Erdélyi**

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

Email:
terdelyi@math.tamu.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-06021-4

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