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Mat. Zametki, 1973, Volume 14, Issue 1, Pages 3–10 (Mi mz7197)  

This article is cited in 2 scientific papers (total in 2 papers)

Optimal rate of integration and $\varepsilon$-entropy of a class of analytic functions

B. D. Boyanovab

a M. V. Lomonosov Moscow State University
b Sofia University

Abstract: The author considers a class $F$ of analytic functions real in the interval $[-1,1]$ and bounded in the unit circle. As an estimate of the optimal quadrature error $R(n)$ over the class $F$ it is shown that
$$ e^{(-2\sqrt2+\frac1{\sqrt2})\pi\sqrt n}\le R(n)\le e{-\frac\pi{\sqrt2}n}. $$
With the additional condition that $\max\limits_{x\in[-1,1]}|f(x)|\le B$, an estimate is obtained for the $\varepsilon$-entropy $H_\varepsilon(F)$:
$$ \frac8{27}\frac{(\ln2)^2}{\pi^2}\le\lim\frac{H_\varepsilon(F)}{(\log\frac1\varepsilon)^3}\le\frac2{\pi^2}(\ln2)^2. $$


Full text: PDF file (458 kB)

English version:
Mathematical Notes, 1973, 14:1, 551–556

Bibliographic databases:

UDC: 517.5
Received: 27.12.1972

Citation: B. D. Boyanov, “Optimal rate of integration and $\varepsilon$-entropy of a class of analytic functions”, Mat. Zametki, 14:1 (1973), 3–10; Math. Notes, 14:1 (1973), 551–556

Citation in format AMSBIB
\Bibitem{Boy73}
\by B.~D.~Boyanov
\paper Optimal rate of integration and $\varepsilon$-entropy of a~class of analytic functions
\jour Mat. Zametki
\yr 1973
\vol 14
\issue 1
\pages 3--10
\mathnet{http://mi.mathnet.ru/mz7197}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=342930}
\zmath{https://zbmath.org/?q=an:0297.65019}
\transl
\jour Math. Notes
\yr 1973
\vol 14
\issue 1
\pages 551--556
\crossref{https://doi.org/10.1007/BF01095768}


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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. K. Yu. Osipenko, “Best methods for approximating analytic functions given with an error”, Math. USSR-Sb., 46:3 (1983), 353–374  mathnet  crossref  mathscinet  zmath
    2. B. D. Boyanov, “Optimal quadrature formulae”, Russian Math. Surveys, 60:6 (2005), 1035–1055  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  • Математические заметки Mathematical Notes
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