The order of comonotone approximation of differentiable periodic functions

Abstract

Let $\Dely$ be a set of all $2\pi$-periodic functions $f$ that are continuous on the real axis $R$\ and\ change their monotonicity at various fixed points $y_{i}\in\lbrack-\pi,\pi),\ i=1,...,2s,\ s\in N$ (i.e., there is a set $Y:=\{y_{i}\}_{i\in\mathbb{Z}}$ of points $y_{i}=y_{i+2s}+2\pi$ on $R$ such that $f$ are nondecreasing on $[y_{i},y_{i-1}]$ if $i$ is even, and nonincreasing if $i$ is odd). In the article, a function $f_{Y}=f\in C^{(1)}\cap\Dely$ has been constructed such that \[ \lim_{n\rightarrow\infty}\sup\frac{n\,E_{n}^{(1)}(f)}{\omega_{4}(f^{\prime },\pi/n)}=\infty, \] where $E_{n}^{(1)}(f)$ is the error of the best uniform approximation of the function $f\in\Dely$ by trigonometric polynomials of order $n\in N$, which also belong to the set $\Dely$, and $\omega_{4}(f^{\prime},\cdot)$ is the $4$-th modulus of smoothness of the function $f^{\prime}.$ So, for a certain constant $c$, the inequality $E_{n}^{(1)}(f)\leq\frac{c}{n}\omega _{3}(f^{\prime},\pi/n)$ is the best with respect to the order of the modulus of smoothness.