Geodesic
Approximation of monotone functions by monotone polynomials
Sbornik. Mathematics, Tome 76 (1993) no. 1, pp. 51-64 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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     title = {Approximation of monotone functions by monotone polynomials},
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     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1993_76_1_a3/}
}
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I. A. Shevchuk. Approximation of monotone functions by monotone polynomials. Sbornik. Mathematics, Tome 76 (1993) no. 1, pp. 51-64. http://geodesic.mathdoc.fr/item/SM_1993_76_1_a3/

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