A~sharp lower bound for the sum of a~sine series with convex coefficients
Sbornik. Mathematics, Tome 207 (2016) no. 12, pp. 1743-1777
Voir la notice de l'article provenant de la source Math-Net.Ru
The sum of a sine series $g(\mathbf b,x)=\sum_{k=1}^\infty b_k\sin kx$ with coefficients forming a convex sequence $\mathbf b$ is known to be positive on the interval $(0,\pi)$. Its values near zero are conventionally evaluated using the Salem function $v(\mathbf b,x)=x\sum_{k=1}^{m(x)} kb_k$, $m(x)=[\pi/x]$. In this paper it is proved that $2\pi^{-2}v(\mathbf b,x)$ is not a minorant for $g(\mathbf b,x)$. The modified Salem function $v_0(\mathbf b,x)=x\bigl(\sum_{k=1}^{m(x)-1} kb_k+(1/2)m(x)b_{m(x)}\bigr)$ is shown to satisfy the lower bound $g(\mathbf b,x)>2\pi^{-2}v_0(\mathbf b,x)$ in some right neighbourhood of zero. This estimate is shown to be sharp on the class of convex sequences $\mathbf b$. Moreover, the upper bound for $g(\mathbf b,x)$ is refined on the class of monotone sequences $\mathbf b$.
Bibliography: 11 titles.
Keywords:
sine series with monotone coefficients, sine series with convex coefficients.
@article{SM_2016_207_12_a6,
author = {A. P. Solodov},
title = {A~sharp lower bound for the sum of a~sine series with convex coefficients},
journal = {Sbornik. Mathematics},
pages = {1743--1777},
publisher = {mathdoc},
volume = {207},
number = {12},
year = {2016},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2016_207_12_a6/}
}
A. P. Solodov. A~sharp lower bound for the sum of a~sine series with convex coefficients. Sbornik. Mathematics, Tome 207 (2016) no. 12, pp. 1743-1777. http://geodesic.mathdoc.fr/item/SM_2016_207_12_a6/