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Mat. Sb., 1992, Volume 183, Number 8, Pages 85–118 (Mi msb1064)  

This article is cited in 18 scientific papers (total in 18 papers)

Best uniform rational approximation of $|x|$ on $[-1,1]$

H. Stahl


Abstract: We consider best rational approximants in the uniform norm to the function $|x|$ on $[-1,1]$. The main result is a proof of a conjecture by R. S. Varga, A. Ruttan, and A. J. Carpenter. They have conjectured that if $E_{nn}(|x|,[-1,1])$, $n\in\mathbb{N}$, denotes the error of the $n$th degree rational approximant, then
\begin{equation} \lim_{n\to\infty}e^{\pi\sqrt n}E_{nn}(|x|,[-1,1])=8. \tag{1} \end{equation}
This conjecture generalizes earlier results, among them most prominently results by D. J. Newman and by N. S. Vyacheslavov.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1993, 76:2, 461–487

Bibliographic databases:

UDC: 517.5
MSC: Primary 41A20; Secondary 41A25
Received: 01.06.1991

Citation: H. Stahl, “Best uniform rational approximation of $|x|$ on $[-1,1]$”, Mat. Sb., 183:8 (1992), 85–118; Russian Acad. Sci. Sb. Math., 76:2 (1993), 461–487

Citation in format AMSBIB
\Bibitem{Sta92}
\by H.~Stahl
\paper Best uniform rational approximation of~$|x|$ on~$[-1,1]$
\jour Mat. Sb.
\yr 1992
\vol 183
\issue 8
\pages 85--118
\mathnet{http://mi.mathnet.ru/msb1064}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1187250}
\zmath{https://zbmath.org/?q=an:0785.41019}
\transl
\jour Russian Acad. Sci. Sb. Math.
\yr 1993
\vol 76
\issue 2
\pages 461--487
\crossref{https://doi.org/10.1070/SM1993v076n02ABEH003422}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1993MK59600011}


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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. Braess D. Lubinsky D. Saff E., “Behavior of Alternation Points in Best Rational Approximation”, Acta Appl. Math., 33:2-3 (1993), 195–210  crossref  mathscinet  zmath  isi
    2. Stahl H., “Best Uniform Rational Approximation of X-Alpha on [0, 1]”, Bull. Amer. Math. Soc., 28:1 (1993), 116–122  crossref  mathscinet  zmath  isi
    3. Kutzelnigg W., “Theory of the Expansion of Wave-Functions in a Gaussian-Basis”, Int. J. Quantum Chem., 51:6 (1994), 447–463  crossref  isi
    4. Stahl H., “Poles and Zeros of Best Rational Approximants of Vertical-Bar-X-Vertical-Bar”, Constr. Approx., 10:4 (1994), 469–522  crossref  mathscinet  zmath  isi
    5. Braess D., “Asymptotics for the Approximation of Wave-Functions by Exponential-Sums”, J. Approx. Theory, 83:1 (1995), 93–103  crossref  mathscinet  zmath  isi
    6. Amos J. Carpenter, “Scientific computation on some mathematical problems”, Journal of Computational and Applied Mathematics, 66:1-2 (1996), 111  crossref
    7. Saff E. Stahl H., “Ray Sequences of Best Rational Approximants for [X](Alpha)”, Can. J. Math.-J. Can. Math., 49:5 (1997), 1034–1065  crossref  mathscinet  zmath  isi
    8. Brutman L., Passow E., “Rational Interpolation to Vertical Bar X Vertical Bar at the Chebyshev Nodes”, Bull. Aust. Math. Soc., 56:1 (1997), 81–86  crossref  mathscinet  zmath  isi
    9. Brutman L., Passow E., “On Rational Interpolation to Vertical Bar X Vertical Bar”, Constr. Approx., 13:3 (1997), 381–391  crossref  mathscinet  zmath  isi
    10. Brutman L., “On Rational Interpolation to Vertical Bar X Vertical Bar at the Adjusted Chebyshev Nodes”, J. Approx. Theory, 95:1 (1998), 146–152  crossref  mathscinet  zmath  isi
    11. A. I. Aptekarev, “Sharp constants for rational approximations of analytic functions”, Sb. Math., 193:1 (2002), 1–72  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    12. S. P. Suetin, “Padé approximants and efficient analytic continuation of a power series”, Russian Math. Surveys, 57:1 (2002), 43–141  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. Xie T., Zhou S., “The Asymptotic Property of Approximation to Vertical Bar X Vertical Bar by Newman's Rational Operators”, Acta Math. Hung., 103:4 (2004), 313–319  crossref  mathscinet  zmath  isi
    14. Lubinsky D.S., “On the Bernstein Constants of Polynomial Approximation”, Constr. Approx., 25:3 (2007), 303–366  crossref  mathscinet  zmath  isi
    15. G. S. Ragimkhanova, A.-R. K. Ramazanov, “Interpolation chain fraction and two extremal problems on rational approximations to $|x|$”, Russian Math. (Iz. VUZ), 51:2 (2007), 33–43  mathnet  crossref  mathscinet  zmath  elib
    16. Yu. A. Labych, A. P. Starovoitov, “Trigonometric Padé approximants for functions with regularly decreasing Fourier coefficients”, Sb. Math., 200:7 (2009), 1051–1074  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    17. Yu. A. Labych, A. P. Starovoitov, “Priblizhenie nepreryvnykh funktsii ratsionalnymi drobyami Pade–Chebysheva”, PFMT, 2011, no. 1(6), 69–78  mathnet
    18. E. A. Rakhmanov, “The Gonchar-Stahl $\rho^2$-theorem and associated directions in the theory of rational approximations of analytic functions”, Sb. Math., 207:9 (2016), 1236–1266  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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