Jackson's type estimate of nearly coconvex approximation

Abstract

Suppose that a continuous on the real axis $2\pi$-periodic function $f$ changes its convexity at $2s,\ s\in\Bbb N,$ points $y_i$ on each period: $-\pi\le y_{2s}<y_{2s-1}<...<y_1<\pi,$ and for the rest $i\in\Bbb Z,$ the points $y_i$ are defined periodically. In the paper, for each $n\ge N,$ a trigonometric polynomial $P_n$ of order $cn$ is found such that: $P_n$ has the same convexity as $f,$ everywhere except, perhaps, the small neighborhoods of the $y_i:$ $$ (y_i-\pi/n,y_i+\pi/n) $$ and $$ \|f-P_n\|\le c(s)\,\omega_4(f,\pi/n), $$ where $N$ is a constant depending only on $\min\limits_{i=1,...,2s}\{y_i-y_{i+1}\},\ c$ and $c(s)$ are constants depending only on $s,\ \omega_4(f,\cdot)$ is the modulus of continuity of the $4$-th order of the function $f,$ and $\|\cdot\|$ is the max-norm.