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Mat. Sb. (N.S.), 1982, Volume 117(159), Number 2, Pages 279–285 (Mi msb2204)  

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

Widths of a class of analytic functions

O. G. Parfenov


Abstract: Let $H^2$ be the Hardy class in the unit disk, $T_r$ the circle of radius $r$ ($0<r<1$) about zero, and $\alpha$ a finite Borel measure on $ T_r$. Denote by $d_n(\alpha)$ the Kolmogorov $n$th-width of the unit ball of $ H^2$ in the metric of $L_2(T_r,\alpha)$. It is proved that
$$ \lim_{n\to\infty}d_n(\alpha)r^{\frac1{2}-n}=\sqrt{g(\alpha)}, $$
where $g(\alpha)$ is the geometric mean of $\alpha$ over $T_r$. The exact values of the diameters are computed for certain measures.
Bibliography: 6 titles.

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English version:
Mathematics of the USSR-Sbornik, 1983, 45:2, 283–289

Bibliographic databases:

UDC: 517.43
MSC: Primary 30D55; Secondary 30D50, 41A60, 46C05, 47B35
Received: 12.11.1980

Citation: O. G. Parfenov, “Widths of a class of analytic functions”, Mat. Sb. (N.S.), 117(159):2 (1982), 279–285; Math. USSR-Sb., 45:2 (1983), 283–289

Citation in format AMSBIB
\Bibitem{Par82}
\by O.~G.~Parfenov
\paper Widths of a class of analytic functions
\jour Mat. Sb. (N.S.)
\yr 1982
\vol 117(159)
\issue 2
\pages 279--285
\mathnet{http://mi.mathnet.ru/msb2204}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=644774}
\zmath{https://zbmath.org/?q=an:0509.30024|0488.30027}
\transl
\jour Math. USSR-Sb.
\yr 1983
\vol 45
\issue 2
\pages 283--289
\crossref{https://doi.org/10.1070/SM1983v045n02ABEH002600}


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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. O. G. Parfenov, “Estimates of the singular numbers of the Carleson imbedding operator”, Math. USSR-Sb., 59:2 (1988), 497–514  mathnet  crossref  mathscinet  zmath
    2. O. G. Parfenov, “Singular numbers of a weighted convolution operator”, Math. USSR-Sb., 61:2 (1988), 309–319  mathnet  crossref  mathscinet  zmath
    3. 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
    4. O. G. Parfenov, “Bernstein widths for some classes of analytic functions”, Math. Notes, 56:4 (1994), 1075–1081  mathnet  crossref  mathscinet  zmath  isi
    5. O. G. Parfenov, “Gel'fand widths of certain classes of analytic functions”, Russian Acad. Sci. Sb. Math., 82:1 (1995), 223–229  mathnet  crossref  mathscinet  zmath  isi
    6. O. G. Parfenov, “Asymptotic Behavior of the Gelfand and the Bernstein Widths of Some Classes of Analytic Functions”, Funct. Anal. Appl., 30:1 (1996), 61–62  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. Baratchart L. Prokhorov V. Saff E., “Best Meromorphic Approximation of Markov Functions on the Unit Circle”, Found. Comput. Math., 1:4 (2001), 385–416  crossref  mathscinet  zmath  isi
    8. Laurent Baratchart, Vasiliy A. Prokhorov, Edward B. Saff, “Asymptotics for Minimal Blaschke Products and Best L1 Meromorphic Approximants of Markov Functions”, Comput. Methods Funct. Theory, 1:2 (2002), 501  crossref  mathscinet
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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