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Mat. Sb. (N.S.), 1987, Volume 133(175), Number 1(5), Pages 86–102 (Mi msb1915)  

This article is cited in 8 scientific papers (total in 8 papers)

Tchebycheff rational approximation in the disk, on the circle, and on a closed interval

A. A. Pekarskii


Abstract: Suppose that the function $f$ is analytic in the disk $ż:|z|<1\}$ and continuous in its closure. Let $R_n(f)$ denote the best uniform approximation of $f$ by rational functions of degree at most $n$. In 1965 Dolzhenko established that if $\sum R_n(f)<\infty$ then $f'$ belongs to the Hardy space $H_1$. The following converse of this result is obtained here: if $f'\in H_1$, then $R_n(f)=O(1/n)$. In combination with results of Peller, Semmes, and the author, this estimate yields, in particular, a description of the set of functions $f$ with $[\sum(2^{k\alpha }R_{2^k}(f))^q]^{1/q}<\infty$, where $\alpha>1$ and $0<q\le\infty$.
Bibliography: 38 titles.

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English version:
Mathematics of the USSR-Sbornik, 1988, 61:1, 87–102

Bibliographic databases:

UDC: 517.53
MSC: Primary 41A20, 41A50; Secondary 30C15, 30D55, 41A25
Received: 01.04.1986

Citation: A. A. Pekarskii, “Tchebycheff rational approximation in the disk, on the circle, and on a closed interval”, Mat. Sb. (N.S.), 133(175):1(5) (1987), 86–102; Math. USSR-Sb., 61:1 (1988), 87–102

Citation in format AMSBIB
\Bibitem{Pek87}
\by A.~A.~Pekarskii
\paper Tchebycheff rational approximation in the disk, on the circle, and on a closed interval
\jour Mat. Sb. (N.S.)
\yr 1987
\vol 133(175)
\issue 1(5)
\pages 86--102
\mathnet{http://mi.mathnet.ru/msb1915}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=899000}
\zmath{https://zbmath.org/?q=an:0656.30031|0631.30035}
\transl
\jour Math. USSR-Sb.
\yr 1988
\vol 61
\issue 1
\pages 87--102
\crossref{https://doi.org/10.1070/SM1988v061n01ABEH003193}


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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. A. Pekarskii, “Uniform rational approximations and Hardy–Sobolev spaces”, Math. Notes, 56:4 (1994), 1082–1088  mathnet  crossref  mathscinet  zmath  isi
    2. Rovba E. Rusak V., “On Approximation Rate by Interpolating Rational Operators with Ordered Poles”, Dokl. Akad. Nauk Belarusi, 41:6 (1997), 21–24  mathscinet  zmath  isi
    3. Rovba E., “On the Approximation of Functions of a Limited Variation by the Freyer and Jackson Rational Operators”, Dokl. Akad. Nauk Belarusi, 42:4 (1998), 13–17  mathscinet  zmath  isi
    4. V. N. Rusak, I. V. Rybachenko, “The Properties of Functions and Approximation by Summation Rational Operators on the Real Axis”, Math. Notes, 76:1 (2004), 103–110  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. A. A. Pekarskii, “Best Uniform Rational Approximations of Functions by Orthoprojections”, Math. Notes, 76:2 (2004), 200–208  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. V. L. Kreptogorskii, “Interpolation of Rational Approximation Spaces Belonging to the Besov Class”, Math. Notes, 77:6 (2005), 809–816  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. A. A. Pekarskii, “Conjugate functions and their connection with uniform rational and piecewise-polynomial approximations”, Sb. Math., 206:2 (2015), 333–340  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. T. S. Mardvilko, A. A. Pekarskii, “Conjugate Functions on the Closed Interval and Their Relationship with Uniform Rational and Piecewise Polynomial Approximations”, Math. Notes, 99:2 (2016), 272–283  mathnet  crossref  crossref  mathscinet  isi  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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