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

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

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

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

Bibliographic databases:

UDC: 517.5
Received: 28.09.2005

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
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\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
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\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}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-48849095644}


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    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  crossref  mathscinet  zmath  isi  elib  scopus
  • Математические заметки Mathematical Notes
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