Geodesic
On the approximation of $x^\alpha$ by rational functions
Izvestiya. Mathematics , Tome 16 (1981) no. 1, pp. 83-101.

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Weak equivalence type estimates are obtained for the smallest deviation of $x^\alpha$ ($\alpha$ a proper fraction) from the rational functions of degree not greater than $n=1,2,\dots$ in the metrics of $L_p[0,1]$ ($1\leqslant p\leqslant\infty$). Bibliography: 15 titles. 
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N. S. Vyacheslavov. On the approximation of $x^\alpha$ by rational functions. Izvestiya. Mathematics , Tome 16 (1981) no. 1, pp. 83-101. http://geodesic.mathdoc.fr/item/IM2_1981_16_1_a4/

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[7] Gonchar A. A., “Skorost ratsionalnoi approksimatsii i svoistvo odnoznachnosti analiticheskoi funktsii v okrestnosti izolirovannoi osoboi tochki”, Matem. sb., 94:2 (1974), 265–282 | MR | Zbl

[8] Vyacheslavov N. S., “O priblizhenii funktsii $|x|$ ratsionalnymi funktsiyami”, Matem. zametki, 16:1 (1974), 163–171 | Zbl

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

[10] Tzimbalario J., “Rational approximation to $x^\alpha$”, J. of Approxim. Theory, 16:2 (1976), 187–193 | DOI | MR | Zbl

[11] Vyacheslavov N. S., “O naimenshikh ukloneniyakh funktsii $\text{sign} x$ i ee pervoobraznykh ot ratsionalnykh funktsii v metrikakh $L_p$, $0

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[12] Cheney E. W., Goldstein A. A., “Mean-square approximation by generalized rational functions”, Math. Zeitschr., 95:3 (1967), 232–241 | DOI | MR | Zbl

[13] Akhiezer N. I., Lektsii po teorii approksimatsii, Nauka, M., 1965 | MR

[14] Gradshtein I. S., Ryzhik I. M., Tablitsy integralov, summ, ryadov i proizvedenii, Nauka, M., 1971

[15] Ramazanov A. K., Priblizhenie funktsii $x^\alpha$ ratsionalnymi drobyami, Tezisy konferentsii “Molodezh i obschestvennyi progress”, Makhachkala, 1978