One counterexample of shape-preserving approximation

M. G. Pleshakov ; S. V. Tyshkevich

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 2, pp. 144-150

Résumé

Let $2s$ points $y_i=-\pi\le y_{2s}\ldots$ 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 $f$.