Geodesic
Asymptotic behavior of the least deviations of the function $\operatorname{sgn}x$ from rational functions
Sbornik. Mathematics, Tome 25 (1975) no. 2, pp. 159-176

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It is shown that the best uniform approximation of $\operatorname{sgn}x$ by rational functions of order at most $n$ on the union of the two intervals $[-1,-\delta]\cup[\delta,1]$ ($0\delta1$) does not exceed $$ e^{\frac{\pi^2}2}\exp\biggl\{-\frac{\pi^2}2\frac n{\ln\frac1\delta+2\ln\ln\frac e\delta+2}\biggr\}. $$ Bibliography: 10 titles. 
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     author = {A. P. Bulanov},
     title = {Asymptotic behavior of the least deviations of the function $\operatorname{sgn}x$ from rational functions},
     journal = {Sbornik. Mathematics},
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     volume = {25},
     number = {2},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1975_25_2_a0/}
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A. P. Bulanov. Asymptotic behavior of the least deviations of the function $\operatorname{sgn}x$ from rational functions. Sbornik. Mathematics, Tome 25 (1975) no. 2, pp. 159-176. http://geodesic.mathdoc.fr/item/SM_1975_25_2_a0/