Uniform estimates of the coconvex approximation of functions by polynomials (Q1311093)

Let \(\overset\circ W^ r\) be the class of continuous functions \(f\) on \(I=[-1,1]\) that have the absolutely continuous \((r-1)\)-th derivative locally in \((-1,1)\), and \(| f^{(r)} (x) (1-x^ 2)^{r/2} | \leq 1\) for almost all \(x\in I\). Theorem: Let \(r \in N\), \(r \geq 3\), \(r\neq 4\). If a function \(f\) is convex on \(I\), and \(f \in {\overset \circ W}^ r\), then for any natural number \(n \geq r-1\) there exists an algebraic polynomial \(P_ n\) of degree \(\leq n\) that is convex on \(I\), and such that, \(| f(x)-P_ n(x) | \leq Cn^{-r}\), \(C=C(r)\), \(x\in I\). This theorem for \(r=1,2\) is a consequence of one in the paper [\textit{D. Leviatan}, Proc. Am. Soc. 98, 471-474 (1986; Zbl 0617.41009)]. The author also proves that the theorem is not true for \(r=4\). A similar theorem for the comonotone approximation in the case \(r=1,2\) follows also from results in the paper by Leviatan, and in the case \(r\geq 3\) it was proved by Dzyubenko, Listopad, and Shevchuk.