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 Mat. Sb. (N.S.), 1982, Volume 117(159), Number 1, Pages 114–130 (Mi msb2185)

Rational approximations of absolutely continuous functions with derivative in an Orlicz space

A. A. Pekarskii

Abstract: Let $R_n(f)$ be the best uniform approximation of $f \in C[0,1]$ by rational fractions of degree at most $n$, and let $W[0,1]$ be the set of monotone convex functions $w\in C[0,1]$ such that $w(0)=0$ and $w(1)=1$.
Theorem 1. Suppose the function $f$ is absolutely continuous on the interval $[0,1],$ and let $w\in W[0,1]$ and $\widehat f= f(w(x))$. If $|\widehat f'|\ln^+|\widehat f'|$ is summable on $[0,1],$ then $R_n(f)=o(1/n)$.
Various applications and generalizations of this result are given, and the periodic case is also considered.
Bibliography: 23 titles.

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English version:
Mathematics of the USSR-Sbornik, 1983, 45:1, 121–137

Bibliographic databases:

UDC: 517.5
MSC: Primary 26A46, 41A20, 46E30; Secondary 41A50

Citation: A. A. Pekarskii, “Rational approximations of absolutely continuous functions with derivative in an Orlicz space”, Mat. Sb. (N.S.), 117(159):1 (1982), 114–130; Math. USSR-Sb., 45:1 (1983), 121–137

Citation in format AMSBIB
\Bibitem{Pek82}
\by A.~A.~Pekarskii
\paper Rational approximations of absolutely continuous functions with derivative in an Orlicz space
\jour Mat. Sb. (N.S.)
\yr 1982
\vol 117(159)
\issue 1
\pages 114--130
\mathnet{http://mi.mathnet.ru/msb2185}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=642493}
\zmath{https://zbmath.org/?q=an:0525.41015}
\transl
\jour Math. USSR-Sb.
\yr 1983
\vol 45
\issue 1
\pages 121--137
\crossref{https://doi.org/10.1070/SM1983v045n01ABEH002590}

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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. Starovoitov A., “Rational Approximation of Functions with a Derivative with a Finite Variation”, Dokl. Akad. Nauk Belarusi, 28:2 (1984), 104–106
2. Devore R., “Approximation by Rational Functions”, Proc. Amer. Math. Soc., 98:4 (1986), 601–604
3. A. A. Pekarskii, “Tchebycheff rational approximation in the disk, on the circle, and on a closed interval”, Math. USSR-Sb., 61:1 (1988), 87–102
4. Pekarskii A., “Direct and Inverse-Theorems of the Rational Approximation and Differential Properties of the Functions”, Dokl. Akad. Nauk Belarusi, 31:6 (1987), 500–503
5. Pekarskii A., “Direct and Converse Theorems of Rational Approximation in the Spaces Lp[-1,1] and C[-1,1]”, 293, no. 6, 1987, 1307–1310
6. Moskona E., Petrushev P., “Characterization of the Rational Approximation in Uniform Metrics”, 42, no. 2, 1989, 37–40
7. Moskona E., Petrushey P., “Uniform Rational Approximation of Functions with 1st Derivative in the Real Hardy Space Re H1”, Constr. Approx., 7:1 (1991), 69–103
8. A. A. Pekarskii, “Uniform rational approximations and Hardy–Sobolev spaces”, Math. Notes, 56:4 (1994), 1082–1088
9. 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
10. Jafarov S.Z., “Approximation of Conjugate Functions by Trigonometric Polynomials in Weighted Orlicz Spaces”, J. Math. Inequal., 7:2 (2013), 271–281
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