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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 $y_i\in[-\pi,\pi)$, $i=1,\dots,2s$, $s\in\mathbb N$. 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
$$
\|f-P_n\|\le c(s)\,\omega_2\biggl(f,\frac\pi n\biggr),
$$
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.
Received: 08.11.2006
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
Linking options:
https://www.mathnet.ru/eng/mzm6357https://doi.org/10.4213/mzm6357 https://www.mathnet.ru/eng/mzm/v84/i5/p713
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Abstract page: | 370 | Full-text PDF : | 155 | References: | 43 | First page: | 4 |
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