Almost coconvex approximation of continuous periodic functions - Dzyubenko, H. A. (Ukrains’kyi Matematychnyi Zhurnal, 2019)
If a $2π$ -periodic function $f$ continuous on the real axis changes its convexity at $2s, s \in N$, points $y_i : π ≤ y_{2s} < y_{2s-1} < . . . < y_1 < π$ , and, for all other $i \in Z$, $y_i$ are periodically defined, then, for every natural $n ≥ N_{y_i}}$, 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 $y_i : (y_i \p_i /n, y_i + π /n)$, and$\| f P_n\| ≤ c(s) ω 4(f, π /n)$,,where $N_{y_i}}$ is a constant depending only on $min_{i = 1,...,2s}\{ y_i y_{i+1}\} , c$ and $c(s)$ are constants depending only on $s, ω 4(f, · )$ is the fourth modulus of smoothness of the function $f$, and $\| · \|$ is the max-norm.