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 Mat. Sb., 2016, Volume 207, Number 12, Pages 124–158 (Mi msb8633)

A sharp lower bound for the sum of a sine series with convex coefficients

A. P. Solodov

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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 $v_0(\mathbf b,x)=x(\sum_{k=1}^{m(x)-1} kb_k+(1/2)m(x)b_{m(x)})$ 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.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-00417-à This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 14-01-00417-a).

DOI: https://doi.org/10.4213/sm8633

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English version:
Sbornik: Mathematics, 2016, 207:12, 1743–1777

Bibliographic databases:

UDC: 517.518.4
MSC: 40A25, 42A32

Citation: A. P. Solodov, “A sharp lower bound for the sum of a sine series with convex coefficients”, Mat. Sb., 207:12 (2016), 124–158; Sb. Math., 207:12 (2016), 1743–1777

Citation in format AMSBIB
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This publication is cited in the following articles:
1. A. Yu. Popov, A. P. Solodov, “Estimates with Sharp Constants of the Sums of Sine Series with Monotone Coefficients of Certain Classes in Terms of the Salem Majorant”, Math. Notes, 104:5 (2018), 702–711
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