Geodesic
On the least deviations of the function $\operatorname{sign}x$ and its primitives from the rational functions in the $L_p$ metrics, $0$
Sbornik. Mathematics, Tome 32 (1977) no. 1, pp. 19-31 
N. S. Vyacheslavov. On the least deviations of the function $\operatorname{sign}x$ and its primitives from the rational functions in the $L_p$ metrics, $0
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     title = {On the least deviations of the function $\operatorname{sign}x$ and its primitives from the rational functions in the $L_p$~metrics, $0<p\leqslant\infty$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper estimates of weak equivalence type, as $n\to\infty$ are given for the least deviations $L_pR_n(f,[-1,1])$ of the functions $f(x)=x^s\operatorname{sign}x$ ($s=0,1,\dots$) in the metric of $L_p[-1,1]$ ($1\leqslant p\leqslant\infty$) from the rational functions of degree $\leqslant n$ ($n=1,2,\dots$). Specifically it is shown that $$ L_pR_n(x^s\operatorname{sign}x,[-1,1])\asymp n^\frac1{2p}\exp\Biggl\{-\pi\sqrt{\biggl(s+\frac1p\biggr)n}\Biggr\} $$ ($s\ne0$ при $p=\infty$); in particular, \begin{gather*} L_pR_n(\operatorname{sign}x,[-1,1])\asymp n^\frac1{2p}\exp\Biggl\{-\pi\sqrt{\frac np}\Biggr\}\qquad(1\leqslant p<\infty), \\ L_pR_n(|x|,[-1,1])\asymp n^\frac1{2p}\exp\Biggl\{-\pi\sqrt{\biggl(1+\frac1p\biggr)n}\Biggr\}\qquad(1\leqslant p\leqslant\infty). \end{gather*} Bibliography: 9 titles.

[1] E. I. Zolotarev, Prilozheniya ellipticheskikh funktsii k voprosam o funktsiyakh, naimenee i naibolee uklonyayuschikhsya ot nulya, Sobr. soch., t. 2, 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] D. J. Newman, “Rational approximation to $|x|$”, Michigan Math. J., 11:1 (1964), 11–14 | DOI | MR | Zbl

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

[5] A. A. Gonchar, “Skorost ratsionalnoi approksimatsii i svoistvo odnoznachnosti analiticheskoi funktsii v okrestnosti izolirovannoi osoboi tochki”, Matem. sb., 94(136) (1974), 265–282 | Zbl

[6] A. P. Bulanov, “Asimptotika dlya naimenshikh uklonenii funktsii $\mathrm{sign}\,x$ ot ratsionalnykh funktsii”, Matem. sb., 96 (138) (1975), 171–188 | MR | Zbl

[7] A. P. Bulanov, “Asimptotika dlya naimenshikh uklonenii $|x|$ ot ratsionalnykh funktsii”, Matem. sb., 76 (118) (1968), 288–303 | MR | Zbl

[8] G. G. Khardi, D. E. Littlvud, G. Polia, Neravenstva, IL, Moskva, 1948

[9] N. S. Vyacheslavov, “O ravnomernom priblizhenii $|x|$ ratsionalnymi funktsiyami”, DAN SSSR, 220:3 (1975), 512–515 | Zbl