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. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 2008, Volume 83, Issue 2, Pages 199–209 (Mi mz4416)

Comonotone Approximation of Periodic Functions

G. A. Dzyubenkoa, M. G. Pleshakovb

a International Mathematical Centre
b Saratov State University named after N. G. Chernyshevsky

Abstract: Suppose that a continuous $2\pi$-periodic function $f$ on the real axis $\mathbb R$ changes its monotonicity at different ordered fixed points $y_i\in [-\pi,\pi)$, $i=1,…,2s$, $s\in\mathbb N$. In other words, there is a set $Y:=\{y_i\}_{i\in\mathbb Z}$ of points $y_i=y_{i+2s}+2\pi$ on $\mathbb R$ such that, on $[y_i,y_{i-1}]$, $f$ is nondecreasing if $i$ is odd and nonincreasing if $i$ is even. For each $n\ge N(Y)$, we construct a trigonometric polynomial $P_n$ of order $\le n$ changing its monotonicity at the same points $y_i\in Y$ as $f$ and such that
$$\|f-P_n\|\le c(s)\omega_2(f,\frac{\pi}{n}),$$
where $N(Y)$ is a constant depending only on $Y$, $c(s)$ is a constant depending only on $s$, $\omega_2(f, \cdot )$ is the modulus of continuity of second order of the function $f$, and $\|\cdot\|$ is the $\max$-norm.

Keywords: $2\pi$-periodic function, comonotone approximation, trigonometric polynomial, Jackson kernel, Whitney's inequality

DOI: https://doi.org/10.4213/mzm4416

Full text: PDF file (508 kB)
References: PDF file   HTML file

English version:
Mathematical Notes, 2008, 83:2, 180–189

Bibliographic databases:

UDC: 517.5

Citation: G. A. Dzyubenko, M. G. Pleshakov, “Comonotone Approximation of Periodic Functions”, Mat. Zametki, 83:2 (2008), 199–209; Math. Notes, 83:2 (2008), 180–189

Citation in format AMSBIB
\Bibitem{DzyPle08} \by G.~A.~Dzyubenko, M.~G.~Pleshakov \paper Comonotone Approximation of Periodic Functions \jour Mat. Zametki \yr 2008 \vol 83 \issue 2 \pages 199--209 \mathnet{http://mi.mathnet.ru/mz4416} \crossref{https://doi.org/10.4213/mzm4416} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2431581} \zmath{https://zbmath.org/?q=an:1151.42001} \transl \jour Math. Notes \yr 2008 \vol 83 \issue 2 \pages 180--189 \crossref{https://doi.org/10.1134/S0001434608010203} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000254056300020} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-48849095644} 

• http://mi.mathnet.ru/eng/mz4416
• https://doi.org/10.4213/mzm4416
• http://mi.mathnet.ru/eng/mz/v83/i2/p199

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles
Cycle of papers

This publication is cited in the following articles:
1. Dzyubenko H. A., “Comonotone approximation of twice differentiable periodic functions”, Ukrainian Math. J., 61:4 (2009), 519–540
•  Number of views: This page: 282 Full text: 96 References: 27 First page: 8