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 Mat. Zametki, 2008, Volume 84, Issue 5, Pages 713–723 (Mi mz6357)

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 $f$ is nondecreasing on $[y_i,y_{i-1}]$ if $i$ is odd and not increasing 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: periodic function, comonotone approximation, trigonometric polynomial, Jackson-type kernel, Whitney's inequality, modulus of continuity

DOI: https://doi.org/10.4213/mzm6357

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English version:
Mathematical Notes, 2008, 84:5, 664–672

Bibliographic databases:

UDC: 517.5

Citation: G. A. Dzyubenko, M. G. Pleshakov, “Comonotone Approximation of Periodic Functions”, Mat. Zametki, 84:5 (2008), 713–723; Math. Notes, 84:5 (2008), 664–672

Citation in format AMSBIB
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