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Mat. Zametki, 1978, Volume 23, Issue 3, Pages 351–360 (Mi mz8150)  

This article is cited in 1 scientific paper (total in 1 paper)

Problem of correctness of the best approximation in the space of continuous functions

A. V. Kolushov

M. V. Lomonosov Moscow State University

Abstract: Let $W^rH_\omega$ the subclass of those functions of $C^r[a,b]$, for which $\omega(f^{(r)},\delta)\le\omega(\delta)$, where $\omega(\delta)$ is a given modulus of continuity, and $P_n$ be the space of algebraic polynomials of degree at most $n$ and $\pi_n(f)$ be the polynomial of best approximation for $f(x)$ on $[a,b]$. Estimates for
$$ A_1(\varepsilon)=\sup_{f\in W^rH_\omega}\sup_{\substack{q_n\in P_n
\|f-q_n\|\le\|f-\pi_n(f)\|+\varepsilon}}\|\pi_n(f)-q_n\|, $$
and moduli of continuity of the operators of best approximation on $W^rH_\omega$ are established. For example, if $\omega(\delta)=\delta^\alpha$, then
\begin{alignat*}{2} A_1(\varepsilon)&\asymp\varepsilon^{(r+\alpha)/(n+r+\alpha)}&&\quadfor \varepsilon<1,
A_1(\varepsilon)&\asymp\varepsilon&&\quadfor \varepsilon>1. \end{alignat*}


Full text: PDF file (591 kB)

English version:
Mathematical Notes, 1978, 23:3, 190–195

Bibliographic databases:

UDC: 517.5
Received: 15.06.1976

Citation: A. V. Kolushov, “Problem of correctness of the best approximation in the space of continuous functions”, Mat. Zametki, 23:3 (1978), 351–360; Math. Notes, 23:3 (1978), 190–195

Citation in format AMSBIB
\Bibitem{Kol78}
\by A.~V.~Kolushov
\paper Problem of correctness of the best approximation in the space of continuous functions
\jour Mat. Zametki
\yr 1978
\vol 23
\issue 3
\pages 351--360
\mathnet{http://mi.mathnet.ru/mz8150}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=493052}
\zmath{https://zbmath.org/?q=an:0423.41013|0401.41032}
\transl
\jour Math. Notes
\yr 1978
\vol 23
\issue 3
\pages 190--195
\crossref{https://doi.org/10.1007/BF01651430}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. P. V. Al'brecht, “Orders of moduli of continuity of operators of almost best approximation”, Russian Acad. Sci. Sb. Math., 83:1 (1995), 1–22  mathnet  crossref  mathscinet  zmath  isi
  • Математические заметки Mathematical Notes
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