Interpolatory pointwise estimates for convex polynomial approximation

Abstract

This paper deals with approximation of smooth convex functions $f$ on an interval by convex algebraic polynomials which interpolate $f$ at the endpoints of this interval. We call such estimates "interpolatory". One important corollary of our main theorem is the following result on approximation of $f\in {\Delta }^{(2)}$, the set of convex functions, from $W^r$, the space of functions on $[-1,1]$ for which $f^{(r-1)}$ is absolutely continuous and $\Vert f^{(r)}{\Vert }_{\infty }:=esssu{p}_{x\in [-1,1]}|f^{(r)}(x)|<\infty$: For any $f\in W^r\cap {\Delta }^{(2)}$, $r\in \mathbb{N}$, there exists a number $\mathcal{N}=\mathcal{N}(f,r)$, such that for every $n\ge \mathcal{N}$, there is an algebraic polynomial of degree $\le n$ which is in ${\Delta }^{(2)}$ and such that \[ \left\| \frac{f-P_n}{φ^r} \right\|_{\infty} \leq \frac{c(r)}{n^r} \left\| f^{(r)}\right\|_{\infty} , \] where $\phi (x):=\sqrt{1-x^2}$. For $r=1$ and $r=2$, the above result holds with $\mathcal{N}=1$ and is well known. For $r\ge 3$, it is not true, in general, with $\mathcal{N}$ independent of $f$.

22 pages. arXiv admin note: substantial text overlap with arXiv:1711.07083