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Fundam. Prikl. Mat., 1995, Volume 1, Issue 3, Pages 581–612 (Mi fpm88)  

This article is cited in 1 scientific paper (total in 1 paper)

New examples of nonnegative trigonometric polynomials with integer coefficients

A. S. Belov

Ivanovo State University

Abstract: In the paper it is proved that for any positive integer $n$ and any number $\lambda\geq1$ the following estimate holds:
$$ 2\lambda n^{\alpha}+\sum_{k=1}^{s}[\lambda(\frac{n}{k})^{\alpha}-1]\cos(kx)>0 $$
for all $x$ and $s=0,\ldots,n$. Here the braces mean the integer part of a number, and $\alpha\in(0,1)$ is the unique root of the equation $\int_{0}^{3\pi/2}t^{-\alpha}\cos t dt=0$. It is proved also that for any positive integer $n$ and any numbers $q\geq2$ and $\lambda \geq sq^q$ the following estimate is true:
$$ 4\lambda n^{1/q}+\sum_{k=1}^{n}[\lambda(( \frac{n}{k})^{1/q}-1)+1]\cos(kx)>0 $$
for all $x$. From these two main results and similar ones new estimates in some extremal problems connected with nonnegative trigonometric polynomials with integer coefficients are deduced.

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Bibliographic databases:
UDC: 517.5
Received: 01.04.1995

Citation: A. S. Belov, “New examples of nonnegative trigonometric polynomials with integer coefficients”, Fundam. Prikl. Mat., 1:3 (1995), 581–612

Citation in format AMSBIB
\Bibitem{Bel95}
\by A.~S.~Belov
\paper New examples of nonnegative trigonometric polynomials with integer coefficients
\jour Fundam. Prikl. Mat.
\yr 1995
\vol 1
\issue 3
\pages 581--612
\mathnet{http://mi.mathnet.ru/fpm88}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1788544}
\zmath{https://zbmath.org/?q=an:0866.42001}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. S. Belov, S. V. Konyagin, “An estimate of the free term of a non-negative trigonometric polynomial with integer coefficients”, Izv. Math., 60:6 (1996), 1123–1182  mathnet  crossref  crossref  mathscinet  zmath  isi
  • Фундаментальная и прикладная математика
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