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 $\|f^{(r)}\|_{\infty} := ess\,sup_{x\in[-1,1]} |f^{(r)}(x)| < \infty$: 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 \[ \left\| \frac{f-P_n}{\varphi^r} \right\|_{\infty} \leq \frac{c(r)}{n^r} \left\| f^{(r)}\right\|_{\infty} , \] where $\varphi(x):= \sqrt{1-x^2}$. 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 $f$.