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Mat. Sb. (N.S.), 1977, Volume 103(145), Number 1(5), Pages 24–36 (Mi msb2709)  

This article is cited in 10 scientific papers (total in 10 papers)

On the least deviations of the function $\operatorname{sign}x$ and its primitives from the rational functions in the $L_p$ metrics, $0<p\leqslant\infty$

N. S. Vyacheslavov


Abstract: In this paper estimates of weak equivalence type, as $n\to\infty$ are given for the least deviations $L_pR_n(f,[-1,1])$ of the functions $f(x)=x^s\operatorname{sign}x$ ($s=0,1,…$) in the metric of $L_p[-1,1]$ ($1\leqslant p\leqslant\infty$) from the rational functions of degree $\leqslant n$ ($n=1,2,…$). Specifically it is shown that
$$ L_pR_n(x^s\operatorname{sign}x,[-1,1])\asymp n^\frac1{2p}\exp\{-\pi\sqrt{(s+\frac1p)n}\} $$
($s\ne0$ при $p=\infty$); in particular,
\begin{gather*} L_pR_n(\operatorname{sign}x,[-1,1])\asymp n^\frac1{2p}\exp\{-\pi\sqrt{\frac np}\}\qquad(1\leqslant p<\infty),
L_pR_n(|x|,[-1,1])\asymp n^\frac1{2p}\exp\{-\pi\sqrt{(1+\frac1p)n}\}\qquad(1\leqslant p\leqslant\infty). \end{gather*}

Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Sbornik, 1977, 32:1, 19–31

Bibliographic databases:

UDC: 517.51
MSC: 41A20
Received: 22.10.1976

Citation: 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, $0<p\leqslant\infty$”, Mat. Sb. (N.S.), 103(145):1(5) (1977), 24–36; Math. USSR-Sb., 32:1 (1977), 19–31

Citation in format AMSBIB
\Bibitem{Vya77}
\by N.~S.~Vyacheslavov
\paper On the least deviations of the function $\operatorname{sign}x$ and its primitives from the rational functions in the $L_p$~metrics, $0<p\leqslant\infty$
\jour Mat. Sb. (N.S.)
\yr 1977
\vol 103(145)
\issue 1(5)
\pages 24--36
\mathnet{http://mi.mathnet.ru/msb2709}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=445174}
\zmath{https://zbmath.org/?q=an:0355.41018|0392.41005}
\transl
\jour Math. USSR-Sb.
\yr 1977
\vol 32
\issue 1
\pages 19--31
\crossref{https://doi.org/10.1070/SM1977v032n01ABEH002313}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1977GE09700002}


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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. N. S. Vyacheslavov, “Rate of approximation of piecewise-analytic functions by rational fractions in the $L_p$-metrics, $0<p\leqslant\infty$”, Math. USSR-Sb., 36:2 (1980), 203–212  mathnet  crossref  mathscinet  zmath  isi
    2. N. S. Vyacheslavov, “On the approximation of $x^\alpha$ by rational functions”, Math. USSR-Izv., 16:1 (1981), 83–101  mathnet  crossref  mathscinet  zmath  isi
    3. Ramazanov A., “Rings of the Coefficients of Rational Functions and Polynomials Best Approximating Functions Chi-Alpha”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1980, no. 5, 41–44  mathscinet  isi
    4. Vjacheslavov N., “Rational Approximation in Weighted Spaces on a Line”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1985, no. 5, 3–10  isi
    5. A. L. Levin, E. B. Saff, “$L_p$ extensions of Gonchar's inequality for rational functions”, Russian Acad. Sci. Sb. Math., 76:1 (1993), 199–210  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. H. Stahl, “Best uniform rational approximation of $|x|$ on $[-1,1]$”, Russian Acad. Sci. Sb. Math., 76:2 (1993), 461–487  mathnet  crossref  mathscinet  zmath  isi
    7. Varga R., “How High-Precision Calculations Can Stimulate Mathematical Research”, Appl. Numer. Math., 10:3-4 (1992), 177–193  crossref  mathscinet  zmath  isi
    8. A.-R. K. Ramazanov, “Rational approximation of functions with finite variation in the Orlicz metric”, Math. Notes, 54:2 (1993), 811–820  mathnet  crossref  mathscinet  zmath  isi
    9. Braess D., “Asymptotics for the Approximation of Wave-Functions by Exponential-Sums”, J. Approx. Theory, 83:1 (1995), 93–103  crossref  mathscinet  zmath  isi
    10. A. K. Ramazanov, “Rational approximation with sign-sensitive weight”, Math. Notes, 60:5 (1996), 536–543  mathnet  crossref  crossref  mathscinet  zmath  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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