@article{Dzyubenko_2019, title={Almost coconvex approximation of continuous periodic functions}, volume={71}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/1444}, abstractNote={If a $2\pi$ -periodic function $f$ continuous on the real axis changes its convexity at $2s, s \in N$, points $y_i : \pi \leq y_{2s} < y_{2s-1} < . . . < y_1 &lt; \pi$ , and, for all other $i \in Z$, $y_i$ are periodically defined, then, for every natural $n \geq N_{y_i }$, we determine a trigonometric polynomial $P_n$ of order cn such that $P_n$ has the same convexity as $f$ everywhere except, possibly, small neighborhoods of the points $y_i : (y_i \p_i /n, y_i + \pi /n)$, and $\| f P_n\| \leq c(s) \omega 4(f, \pi /n)$,, where $N_{y_i }$ is a constant depending only on $\mathrm{m}\mathrm{i}\mathrm{n}_{i = 1,...,2s}\{ y_i y_{i+1}\} , c$ and $c(s)$ are constants depending only on $s, \omega 4(f, \cdot )$ is the fourth modulus of smoothness of the function $f$, and $\| \cdot \|$ is the max-norm.}, number={3}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Dzyubenko, H. A.}, year={2019}, month={Mar.}, pages={353-367} }