Abstract:
It is shown that if $p\geqslant1$, and if the function $f(x)$ has a convex $p$th derivative for $x\in[a,b]$, then the least uniform deviation of $f$ from the rational functions of degree no higher than $n$ is bounded from above by the quantity
$$
C(p,\nu)M(b-a)^pn^{-p-2}\overbrace{\ln\dots\ln}^{\nu\,\text{times}}n
$$
where $\nu$ is a natural number and $C(p,\nu)$ depends only on $p$ and $\nu$, and where $M=\max|f^{(p)}(x)|$. There is an analogous estimate for $p=0$, provided that $f(x)$ is convex and $f\in{\operatorname{Lip}(\alpha)}$ for some $\alpha>0$.
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