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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
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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.

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