One Counterexample of Comonotone Approximation of $2π$-periodic Function on Trigonometric Polynomials

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 $f\in\bigtriangleup^{(1)}(Y)$ 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 $k\in\Bbb N$, $k\>3$, and $n\in\Bbb N$ there a function $f(x):=f(x;s,Y,n,k)$ exists, such that $f\in\bigtriangleup^{(1)}(Y)\bigcap{\Bbb C}^{(1)}$ and $$ E_n^{(1)}(f;Y)>B_Yn^{\frac k3 -1}\frac 1n\omega_k\left(f';\frac 1n\right), $$ where $B_Y=$const, depending only on $Y$ and $k$; $\omega_k$ is the modulus of smoothness of order $k$, of $f$.