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 
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/
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     title = {Asymptotic behavior of the least deviations of the function $\operatorname{sgn}x$ from rational functions},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $[-1,-\delta]\cup[\delta,1]$ ($0<\delta<1$) 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.

[1] E. I. Zolotarev, “Prilozhenie ellipticheskikh funktsii k voprosam o funktsiyakh, naimenee i naibolee uklonyayuschikhsya ot nulya”, Sobr. soch., t. II, izd-vo AN SSSR, Moskva, 1932

[2] N. I. Akhiezer, “Ob odnoi zadache E. I. Zolotareva”, Izv. AN SSSR, seriya matem., 1929, 919–931

[3] N. I. Akhiezer, Lektsii po teorii approksimatsii, izd-vo «Nauka», Moskva, 1965 | MR

[4] N. I. Akhiezer, Elementy teorii ellipticheskikh funktsii, izd-vo «Nauka>, Moskva, 1970 | MR

[5] A. A. Gonchar, “Otsenki rosta ratsionalnykh funktsii i nekotorye ikh prilozheniya”, Matem. sb., 72(114) (1967), 489–503 | Zbl

[6] A. A. Gonchar, “O zadachakh E. I. Zolotareva, svyazannykh s ratsionalnymi funktsiyami”, Matem. sb., 78(120) (1969), 640–654 | Zbl

[7] M. A. Lavrentev, B. V. Shabat, Metody teorii funktsii kompleksnogo peremennogo, izd-vo «Nauka», Moskva, 1965 | MR

[8] I. P. Natanson, Konstruktivnaya teoriya funktsii, Gostekhizdat, Moskva–Leningrad, 1949 | MR

[9] I. S. Gradshtein, I. M. Ryzhik, Tablitsy integralov, summ, ryadov i proizvedenii, Fizmatgiz, Moskva, 1963

[10] G. M. Fikhtengolts, Kurs differentsialnogo i integralnogo ischisleniya, t. II, Fizmatgiz, Moskva, 1962