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This article is cited in 2 scientific papers (total in 2 papers)
Mathematics
One counterexample of shape-preserving approximation
M. G. Pleshakov, S. V. Tyshkevich Saratov State University, 83, Astrakhanskaya str., 410012, Saratov, Russia
Abstract:
Let $2s$ points $y_i=-\pi\le y_{2s}<\ldots<y_1<\pi$ be given. Using these points, we define the points $y_i$ for all integer indices $i$ by the equality $y_i=y_{i+2s}+2\pi$. We shall write $f\in\Delta^{(1)}(Y)$ if $f$ is a $2\pi$-periodic function and $f$ does not decrease on $[y_i,~y_{i-1}]$ if $i$ is odd; and $f$ does not increase on $[y_i,y_{i-1}]$ if $i$ is even. We denote $E_n^{(1)}(f;Y)$ the value of the best uniform comonotone approximation. In this article the following counterexample of comonotone approximation is proved.
Example. For each $k\in\mathbb N$, $k>2$, and $n\in\mathbb N$ there a function $f(x):=f(x;s,Y,n,k)$ exists, such that $f\in\Delta^{(1)}(Y)$ and
$$
E_n^{(1)}(f;Y)>B_Yn^{\frac k2-1}\omega_k\left(f;\frac1n\right),
$$
where $B_Y=\mathrm{cons}$t, depending only on $Y$ and $k$; $\omega_k$ is the modulus of smoothness of order $k$, of $f$.
Key words:
trigonometric polynomials, polynomial approximation, shape-preserving.
Citation:
M. G. Pleshakov, S. V. Tyshkevich, “One counterexample of shape-preserving approximation”, Izv. Saratov Univ. Math. Mech. Inform., 14:2 (2014), 144–150
Linking options:
https://www.mathnet.ru/eng/isu497 https://www.mathnet.ru/eng/isu/v14/i2/p144
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