Abstract:
The following theorem is proved for the case $k+r>2$.
Theorem. If $k$, $r\in{\mathbb N}$, $I:=[-1,1]$, and the function $f=f(x)$ is nondecreasing on $I$ and has $r$ continuous derivatives, then for each positive integer $n\geqslant r + k - 1$ there is an algebraic polynomial $P_n = P_n(x)$ of degree $\leqslant n$ that is nondecreasing on $I$ and such that for all $x\in I$ $$
|f(x)-P_n(x)|\leqslant c\biggl({1\over n^2}+{\sqrt {1-x^2}\over n}\,\biggr)^r
\omega _k\biggl(f^{(r)};{1\over n^2}+{\sqrt{1-x^2}\over n}\,\biggr), \qquad
c=c(r,k),
$$
where $\omega_k(f^{(r)};\,t)$ is the $k$th-order modulus of continuity of the function
$f^{(r)}=f^{(r)}(x)$.
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