This article is cited in 7 papers

An estimate of the free term of a non-negative trigonometric polynomial with integer coefficients

A. S. Belov^a, S. V. Konyagin<sup>b

a</sup> Ivanovo State University
^b M. V. Lomonosov Moscow State University

Abstract: We denote by $M_Z^{\downarrow}(n)$ (resp., $K_Z^{\downarrow}(n)$) the smallest value of $a_0$ that can occur in a non-negative trigonometric polynomial
$$ \sum_{k=0}^n a_k\cos(kx) $$
with non-negative integer coefficients $a_1\geqslant a_2\geqslant\dots\geqslant a_n$ such that $a_n\geqslant 1$ (resp., $\sum_{k=1}^n a_k=n$). We prove that for all natural numbers $n\geqslant 3$
$$ \dfrac{\ln^2 n}{\ln\ln n}\ll K_Z^\downarrow(n)\ll M_Z^\downarrow(n)\ll(\ln n)^3. $$


MSC: 42A05

Received: 05.04.1996

DOI: 10.4213/im95

 English version: Izvestiya: Mathematics, 1996, 60:6, 1123–1182

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