The rational approximation of convex functions of the class $\operatorname{Lip}\alpha$

A. Khatamov

Matematičeskie zametki, Tome 18 (1975) no. 6, pp. 845-854.

Résumé

It is proved that if a function $f(x)$ is convex on $[a,b]$ and $f\in\operatorname{Lip}_{K(f)}\alpha$, $0\alpha1$, then the least uniform deviation of this function from rational functions of degree not higher than $n$ does not exceed $C(\alpha,\nu)(b-a)^\alpha K(f)\cdot n^{-2}\cdot\overbrace{\ln\dots\ln n}^{\nu\text{раз}}$ ($\nu$ is a natural number; $C(\alpha,\nu)$ depends only on $\alpha$ and $\nu$; $K(f)$ is a Lipschitz constant; and $n\ge n(\nu)=\min\{n:\overbrace{\ln\dots\ln n}^{\nu\text{раз}}\}$).