Geodesic
Best Uniform Rational Approximations of Functions by Orthoprojections
Matematičeskie zametki, Tome 76 (2004) no. 2, pp. 216-225.

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Let $C[-1,1]$ be the Banach space of continuous complex functions $f$ on the interval $[-1,1]$ equipped with the standard maximum norm $\|f\|$; let $\omega(\,\cdot\,)=\omega(\,\cdot\,,f$ be the modulus of continuity of $f$; and let $R_n=R_n(f)$ be the best uniform approximation of $f$ by rational functions (r.f.) whose degrees do not exceed $n=1,2,\ldots$. The space $C[-1,1]$ is also regarded as a pre-Hilbert space with respect to the inner product given by $(f,g)=(1/\pi)\int_{-1}^1f(x)g(x)(1-x^2)^{-1/2}\,dx$. Let $z_n=\{z_1,z_2,\ldots,z_n\}$ be a set of points located outside the interval $[-1,1]$. By $F(\,\cdot\,,f,z_n)$ we denote an orthoprojection operator acting from the pre-Hilbert space $C[-1,1]$ onto its $(n+1)$-dimensional subspace consisting of rational functions whose poles (with multiplicity taken into account) can only be points of the set $z_n$. In this paper, we show that if $f$ is not a rational function of degree $\leqslant n$, then we can find a set of points $z_n=z_n(f)$ such that $\|f(\,\cdot\,)-F(\,\cdot\,,f,z_n)\|\leqslant 12R_n\ln\frac3{\omega^{-1}(R_n/3)}$. 
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A. A. Pekarskii. Best Uniform Rational Approximations of Functions by Orthoprojections. Matematičeskie zametki, Tome 76 (2004) no. 2, pp. 216-225. http://geodesic.mathdoc.fr/item/MZM_2004_76_2_a5/

[1] Uolsh Dzh. L., Interpolyatsiya i approksimatsiya ratsionalnymi funktsiyami v kompleksnoi oblasti, IL, M., 1961

[2] Dzhrbashyan M. M., Kitbalyan A. A., “Ob odnom obobschenii polinomov Chebysheva”, Dokl. AN Arm. SSR, 37:5 (1964), 263–270 | MR

[3] Timan A. F., Teoriya priblizheniya funktsii deistvitelnogo peremennogo, GIFML, M., 1960

[4] Rusak V. N., “O skorosti priblizheniya nekotorykh klassov funktsii ratsionalnymi operatorami. Teoriya priblizheniya funktsii”, Trudy Mezhdunarodnoi konferentsii po teorii priblizheniya funktsii (Kiev, 31 maya–5 iyunya, 1983), M., 1987, 382–386

[5] Rusak V. N., Ratsionalnye funktsii kak apparat priblizheniya, Diss. ... dokt. fiz.-mat. nauk, Kiev, 1987

[6] Pekarskii A. A., Rovba E. A., “Ravnomernye priblizheniya funktsii Stiltesa posredstvom ortoproektsii na mnozhestvo ratsionalnykh funktsii”, Matem. zametki, 65:3 (1999), 362–368 | MR | Zbl

[7] Pekarskii A. A., “Chebyshevskie ratsionalnye priblizheniya v kruge na okruzhnosti i na otrezke”, Matem. sb., 133:1 (1987), 86–102 | MR | Zbl

[8] Lorentz G. G., Golitschek M. V., Makovoz Y., Constructive Approximation. Advanced Problems, Springer, Berlin, 1996

[9] Dzhrbashyan M. M., “K teorii ryadov Fure po ratsionalnym funktsiyam”, Izv. AN Arm. SSR. Ser. matem., 9:7 (1956), 3–28 | MR

[10] Pekarskii A. A., “Otsenki proizvodnoi integrala tipa Koshi s meromorfnoi plotnostyu i ikh prilozheniya”, Matem. zametki, 31:3 (1982), 389–402 | MR | Zbl

[11] Ulyanov P. L., “O priblizhenii funktsii”, Sib. matem. zh., 5:2 (1964), 418–437 | MR | Zbl