RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Sb.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

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

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-а
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

Full text: PDF file (669 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 517.518.4
MSC: 40A25, 42A32
Received: 10.11.2015

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
\Bibitem{Sol16}
\by A.~P.~Solodov
\paper A~sharp lower bound for the sum of a~sine series with convex coefficients
\jour Mat. Sb.
\yr 2016
\vol 207
\issue 12
\pages 124--158
\mathnet{http://mi.mathnet.ru/msb8633}
\crossref{https://doi.org/10.4213/sm8633}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3588989}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2016SbMat.207.1743S}
\elib{http://elibrary.ru/item.asp?id=27485046}
\transl
\jour Sb. Math.
\yr 2016
\vol 207
\issue 12
\pages 1743--1777
\crossref{https://doi.org/10.1070/SM8633}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000394542200007}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85014241828}


Linking options:
  • http://mi.mathnet.ru/eng/msb8633
  • https://doi.org/10.4213/sm8633
  • http://mi.mathnet.ru/eng/msb/v207/i12/p124

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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  mathnet  crossref  crossref  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
    Number of views:
    This page:363
    Full text:11
    References:53
    First page:70

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020