Proceedings of the American Mathematical Society

Author(s): Peter Borwein; Tamás Erdélyi
Journal: Proc. Amer. Math. Soc. 129 (2001), 725-730.
MSC (2000): Primary 41A17
Posted: November 3, 2000
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Abstract: We examine the size of a real trigonometric polynomial of degree at most

having at least

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.

[Bec-95]: J. Beck, Flat polynomials on the unit circle - note on a problem of Littlewood, Bull. London Math. Soc. 23 (1991), 269-277. MR 93b:42002
[Bor-98]: P. Borwein, Some Old Problems on Polynomials with Integer Coefficients, in Approximation Theory IX, ed. C. Chui and L. Schumaker, Vanderbilt University Press (1998), 31-50. CMP 2000:11
[Bor-95]: P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 1995. MR 97e:41001
[Er-62]: P. Erdos, An inequality for the maximum of trigonometric polynomials, Annales Polonica Math. 12 (1962), 151-154. MR 25:5330
[Kah-85]: J-P. Kahane, Sur les polynômes á coefficients unimodulaires, Bull. London Math. Soc 12 (1980), 321-342. MR 82a:30003
[Li-61]: J.E. Littlewood, On the mean value of certain trigonometric polynomials, Jour. London Math. Soc. 36 (1961), 307-334. MR 25:5331a
[Li-62]: J.E. Littlewood, On the mean value of certain trigonometric polynomials (II), Jour. London Math. Soc. 39 (1964), 511-552.
[Li-66a]: J.E. Littlewood, The real zeros and value distributions of real trigonometrical polynomials, Jour. London Math. Soc. 41 (1966), 336-342. MR 33:4559
[Li-66b]: J.E. Littlewood, On polynomials $\sum \pm z^{m}$ and $\sum e^{\alpha _{m}i} z^{m}$ , $z=e^{\theta i}$ , Jour. London Math. Soc. 41 (1966), 367-376. MR 33:4237
[Li-68]: J.E. Littlewood, Some Problems in Real and Complex Analysis, Heath Mathematical Monographs, Lexington, Massachusetts, 1968. MR 39:5777
[Saf-90]: B. Saffari, Barker sequences and Littlewood's ``two sided conjectures'' on polynomials with $\pm 1$ coefficients, Séminaire d'Analyse Harmonique, 1989/90, Univ. Paris XI, Orsay, 1990, 139-151. MR 92i:11032

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 41A17

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: 10.1090/S0002-9939-00-06021-4
PII: S 0002-9939(00)06021-4
Keywords: Real trigonometric polynomials, real zeros, unimodular trigonometric polynomials, flatness
Received by editor(s): March 2, 1999
Posted: 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
Copyright of article: Copyright 2000, Copyright retained by the authors

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