Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On nonnegative cosine polynomials with nonnegative integral coefficients


Author: Mihail N. Kolountzakis
Journal: Proc. Amer. Math. Soc. 120 (1994), 157-163
MSC: Primary 42A05; Secondary 42A32
MathSciNet review: 1169037
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that there exist $ {p_0} > 0$ and $ {p_1}, \ldots ,{p_N}$ nonnegative integers, such that

$\displaystyle 0 \leqslant p(x) = {p_0} + {p_1}\cos x + \cdots + {p_N}\cos Nx$

and $ {p_0} \ll {s^{1/3}}$ for $ s = \sum\nolimits_{j = 0}^N {{p_j}} $, improving on a result of Odlyzko who showed the existence of such a polynomial $ p$ that satisfies $ {p_0} \ll {(s\log s)^{1/3}}$. Our result implies an improvement of the best known estimate for a problem of Erdős and Szekeres. As our method is nonconstructive, we also give a method for constructing an infinite family of such polynomials, given one good "seed" polynomial.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42A05, 42A32

Retrieve articles in all journals with MSC: 42A05, 42A32


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1169037-9
PII: S 0002-9939(1994)1169037-9
Article copyright: © Copyright 1994 American Mathematical Society