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 $2s,s\in \mathbb{N},$ points $y_i$ on each period: $-\pi \le y_{2s}<y_{2s-1}<...<y_1<\pi ,$ and for the rest $i\in \mathbb{Z},$ 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 $$\Vert f-P_n\Vert \le c(s){\omega }_4(f,\pi /n),$$ where $N$ is a constant depending only on $\underset{i=1,...,2s}{\min }\{y_i-y_{i+1}\},c$ 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 $\Vert \cdot \Vert$ is the max-norm.