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 Mat. Sb. (N.S.), 1975, Volume 96(138), Number 2, Pages 171–188 (Mi msb3122)

Asymptotic behavior of the least deviations of the function $\operatorname{sgn}x$ from rational functions

A. P. Bulanov

Abstract: 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
$$e^{\frac{\pi^2}2}\exp\{-\frac{\pi^2}2\frac n{\ln\frac1\delta+2\ln\ln\frac e\delta+2}\}.$$

Bibliography: 10 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 25:2, 159–176

Bibliographic databases:

UDC: 517.5
MSC: 41A20, 41A50

Citation: A. P. Bulanov, “Asymptotic behavior of the least deviations of the function $\operatorname{sgn}x$ from rational functions”, Mat. Sb. (N.S.), 96(138):2 (1975), 171–188; Math. USSR-Sb., 25:2 (1975), 159–176

Citation in format AMSBIB
\Bibitem{Bul75}
\by A.~P.~Bulanov
\paper Asymptotic behavior of the least deviations of the function $\operatorname{sgn}x$ from rational functions
\jour Mat. Sb. (N.S.)
\yr 1975
\vol 96(138)
\issue 2
\pages 171--188
\mathnet{http://mi.mathnet.ru/msb3122}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=370006}
\zmath{https://zbmath.org/?q=an:0317.41012}
\transl
\jour Math. USSR-Sb.
\yr 1975
\vol 25
\issue 2
\pages 159--176
\crossref{https://doi.org/10.1070/SM1975v025n02ABEH002204}

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. P. Bulanov, “Rational approximation of continuous functions of finite variation”, Math. USSR-Izv., 9:5 (1975), 1081–1118
2. E. P. Dolzhenko, V. I. Danchenko, “Dependence of the differentiability of functions of several variables on their rate of approximation by rational functions”, Math. USSR-Izv., 11:1 (1977), 171–192
3. 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, $0<p\leqslant\infty$”, Math. USSR-Sb., 32:1 (1977), 19–31
4. A.R Reddy, “Recent advances in Chebyshev rational approximation on finite and infinite intervals”, Journal of Approximation Theory, 22:1 (1978), 59
5. Vladov N., “Exact Asymptotics for the Rational Approximation of Sign(X) in Hausdorff Metric”, 39, no. 7, 1986, 37–40
6. A.-R. K. Ramazanov, “Rational approximation of functions with finite variation in the Orlicz metric”, Math. Notes, 54:2 (1993), 811–820
7. A. P. Bulanov, “Rational approximation of functions of several variables with finite Hardy variation”, Sb. Math., 186:11 (1995), 1599–1620
8. V. I. Danchenko, “Estimates of derivatives of simplest fractions and other questions”, Sb. Math., 197:4 (2006), 505–524
9. V. I. Danchenko, M. A. Komarov, P. V. Chunaev, “Ekstremalnye i approksimativnye svoistva naiprosteishikh drobei”, Izv. vuzov. Matem., 2018, no. 12, 9–49
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