Geodesic
Coincidence of Least Uniform Deviations of Functions from Polynomials and Rational Fractions
Matematičeskie zametki, Tome 74 (2003) no. 4, pp. 612-617.

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For a given nonincreasing vanishing sequence $\{a_n\}^\infty_{n=0}$ of nonnegative real numbers, we find necessary and sufficient conditions for a sequence $\{n_k\}^\infty_{k=0}$ to have the property that for this sequence there exists a function f continuous on the interval $[0,1]$ and satisfying the condition that $R_{n_k,m_k}(f)=E_{n_k}(f)=a_{n_k}$, $k=0,1,2,\dots$, where $E_n(f)$ and $R_{n,m}(f)$ are the best uniform approximations to the function $f$ by polynomials whose degree does not exceed $n$ and by rational functions of the form $r_{n,m}(x)=p_n(x)/q_m(x)$, respectively. 
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A. P. Starovoitov. Coincidence of Least Uniform Deviations of Functions from Polynomials and Rational Fractions. Matematičeskie zametki, Tome 74 (2003) no. 4, pp. 612-617. http://geodesic.mathdoc.fr/item/MZM_2003_74_4_a13/

[1] Daugavet I. K., Vvedenie v teoriyu priblizheniya funktsii, LGU, L., 1977

[2] Dolzhenko E. P., “Sravnenie skorostei ratsionalnoi i polinomialnoi approksimatsii”, Matem. zametki, 1:3 (1967), 313–320 | MR | Zbl

[3] Boehm B., “Functions, whose best rational Chebyshev approximations are polinomials”, Numer. Math., 6:3 (1964), 235–242 | DOI | MR | Zbl

[4] Lorentz G., v. Golitschek M., Makavoz Y., Constructive Approximation. Advanced Problems, Springer-Verlag, New York, 1996

[5] Pekarskii A. A., “O sluchayakh sovpadeniya nailuchshikh ravnomernykh polinomialnykh i ratsionalnykh priblizhenii”, J. Technical University at Plovdiv, 2, Fund. Sci. and Appl. (1996), 41–44 | MR

[6] Nazarenko M. A., Nekotorye svoistva ratsionalnykh approksimatsii., Diss. $\dots$ k.f.-m.n., MGU, M., 1997

[7] Akhiezer N. I., Lektsii po teorii approksimatsii, Nauka, M., 1965

[8] Gonchar A. A., “Obratnye teoremy o nailuchshikh priblizheniyakh na zamknutykh mnozhestvakh”, Dokl. AN SSSR, 128:1 (1959), 25–28 | MR | Zbl

[9] Vyacheslavov N. S., Ramazanov A. K., “O stepeni ratsionalnykh funktsii nailuchshego priblizheniya v $L_p(R^m)$”, Matem. zametki, 53:2 (1993), 37–45 | MR | Zbl

[10] Pekarskii A. A., “Suschestvovanie funktsii s zadannymi nailuchshimi ravnomernymi ratsionalnymi priblizheniyami”, Izv. AN Belarusi. Ser. fiz.-matem., 1994, no. 1, 23–26 | MR | Zbl