Degree of comonotone approximation of periodic functions

Abstract

If a continuously differentiable on a real axe $\Bbb R\ \ 2\pi - $periodic function $f$ changes its monotonicity at different fixed points $y_i\in [-\pi,\pi),\ i=1,...,2s,\ s\in\Bbb N ,$ (i.e., on $\Bbb R$ there is a set $Y:=\{y_i\}_{i\in\Bbb Z}$ of points $y_i=y_{i+2s}+2\pi $ such that on $[y_i,y_{i-1}]\ f$ is nondecreasing if $i$ is odd, and nonincreasing if $i$ is even), then for each natural number $n,\ n\ge N(Y)=const,$ in the article a trigonometric polynomial $T_n $ of order $\le n,$ which changes its monotonicity at the same points $y_i\in Y,$ like $f,$ is found such that $$\left\Vert f-T_n \right \Vert \le \frac{c(s)}{n}\, \omega_3\left(f',1/n\right),$$ where $N(Y)$ depends only on $Y,$ $c(s)-$ constant which is depending only on $s,\ \omega_3 \left(f,\cdot\right)-$ modulus of smoothness of order $3$ of the function $f$ and $\Vert \cdot \Vert -\max $-norm. Also the other estimates that are possible in this kind of approximation are listed.