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 $fin Delta^{(2)}$, the set of convex functions, from $W^r$, the space of functions on $[-1,1]$ for which $f^{(r-1)}$ is absolutely continuous and $|f^{(r)}|_{infty} := ess,sup_{xin[-1,1]} |f^{(r)}(x)| < infty$: For any $fin W^r capDelta^{(2)}$, $rin {mathbb N}$, there exists a number ${mathcal N}={mathcal N}(f,r)$, such that for every $nge {mathcal N}$, 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 $rge 3$, it is not true, in general, with ${mathcal N}$ independent of $f$.
تحميل البحث