Geodesic
Best uniform rational approximation of~$|x|$ on~$[-1,1]$
Sbornik. Mathematics, Tome 76 (1993) no. 2, pp. 461-487

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We consider best rational approximants in the uniform norm to the function $|x|$ on $[-1,1]$. The main result is a proof of a conjecture by R. S. Varga, A. Ruttan, and A. J. Carpenter. They have conjectured that if $E_{nn}(|x|,[-1,1])$, $n\in\mathbb{N}$, denotes the error of the $n$th degree rational approximant, then \begin{equation} \lim_{n\to\infty}e^{\pi\sqrt n}E_{nn}(|x|,[-1,1])=8. \tag{1} \end{equation} This conjecture generalizes earlier results, among them most prominently results by D. J. Newman and by N. S. Vyacheslavov. 
@article{SM_1993_76_2_a10,
     author = {H. Stahl},
     title = {Best uniform rational approximation of~$|x|$ on~$[-1,1]$},
     journal = {Sbornik. Mathematics},
     pages = {461--487},
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     volume = {76},
     number = {2},
     year = {1993},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1993_76_2_a10/}
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H. Stahl. Best uniform rational approximation of~$|x|$ on~$[-1,1]$. Sbornik. Mathematics, Tome 76 (1993) no. 2, pp. 461-487. http://geodesic.mathdoc.fr/item/SM_1993_76_2_a10/