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

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

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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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
2. Rovba E. Rusak V., “On Approximation Rate by Interpolating Rational Operators with Ordered Poles”, Dokl. Akad. Nauk Belarusi, 41:6 (1997), 21–24
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
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
5. A. A. Pekarskii, “Best Uniform Rational Approximations of Functions by Orthoprojections”, Math. Notes, 76:2 (2004), 200–208
6. V. L. Kreptogorskii, “Interpolation of Rational Approximation Spaces Belonging to the Besov Class”, Math. Notes, 77:6 (2005), 809–816
7. A. A. Pekarskii, “Conjugate functions and their connection with uniform rational and piecewise-polynomial approximations”, Sb. Math., 206:2 (2015), 333–340
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
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