(FUNDAMENTAL AND APPLIED MATHEMATICS)

1995, VOLUME 1, NUMBER 3, PAGES 581-612

## New examples of nonnegative trigonometric polynomials with integer coefficients

A.S.Belov

In the paper it is proved that for any positive integer n and any number $\lambda \geq 1$ the following estimate holds:

$2 \lambda n^{\alpha} + \sum_{k=1}^s [ \lambda ( \frac nk )^{\alpha} - 1] \cos (kx) > 0$
for all x and s = 0,..., 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 $q \geq 2$ and $\lambda \geq sq^q$ the following estimate is true:

$4\lambda n^{1/q} + \sum_{k=1}^n [ \lambda (( \frac nk )^{1/q} - 1) + 1 ] \cos (kx) > 0$
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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