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Mat. Sb. (N.S.), 1981, Volume 115(157), Number 4(8), Pages 577–589 (Mi msb2417)  

This article is cited in 3 scientific papers (total in 3 papers)

Coconvex approximation of functions of several variables by polynomials

A. S. Shvedov


Abstract: Let $M\subseteq\mathbf R^m$ be a compact convex body, and $O$ the center of gravity of $M$. For a convex function $f\colon M\to\mathbf R$ let
$$ \omega(f,\delta,M)=\sup_{\substack{x,y\in M
|x-y|_M\leqslant\delta}}|f(x)-f(y)|\qquad(\delta\geqslant0), $$
where $|x|_M=\min\{\mu\geqslant0:x\in\mu(M-O)\}$, and let $M_1\subseteq\mathbf R^m$ be a convex body, $M\subseteq M_1$, and $\varkappa=\min\{\mu\geqslant1:M_1\subseteq\mu M\}$, $\mu M$ being a homothety of $M$ with respect to $O$. Then for $n\geqslant0$ there exists an algebraic polynomial
$$ p_n(x)=\sum_{i_1+…+i_m\leqslant n}a_{i_1,…,i_m}x^{i_1}_1\cdots x^{i_m}_m $$
that is convex on $M_1$ and such that
$$ \|f-p_n\|_{C(M)}\leqslant\varkappa A_m\omega(f,\frac1{n+1},M). $$

Bibliography: 6 titles.

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English version:
Mathematics of the USSR-Sbornik, 1982, 43:4, 515–526

Bibliographic databases:

UDC: 517.5
MSC: Primary 26B25, 41A10, 41A17; Secondary 26A15, 52A40
Received: 29.02.1980

Citation: A. S. Shvedov, “Coconvex approximation of functions of several variables by polynomials”, Mat. Sb. (N.S.), 115(157):4(8) (1981), 577–589; Math. USSR-Sb., 43:4 (1982), 515–526

Citation in format AMSBIB
\Bibitem{Shv81}
\by A.~S.~Shvedov
\paper Coconvex approximation of functions of several variables by polynomials
\jour Mat. Sb. (N.S.)
\yr 1981
\vol 115(157)
\issue 4(8)
\pages 577--589
\mathnet{http://mi.mathnet.ru/msb2417}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=629628}
\zmath{https://zbmath.org/?q=an:0506.41005}
\transl
\jour Math. USSR-Sb.
\yr 1982
\vol 43
\issue 4
\pages 515--526
\crossref{https://doi.org/10.1070/SM1982v043n04ABEH002578}


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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. Kroo A., “On the Approximation of Convex Bodies by Convex Algebraic Level Surfaces”, J. Approx. Theory, 162:3, SI (2010), 628–637  crossref  mathscinet  zmath  isi
    2. Konovalov V.N., Kopotun K.A., Maiorov V.E., “Convex Polynomial and Ridge Approximation of Lipschitz Functions in R-D”, Rocky Mt. J. Math., 40:3 (2010), 957–976  crossref  mathscinet  zmath  isi
    3. Kroo A., “Rate of Approximation of Regular Convex Bodies by Convex Algebraic Level Surfaces”, Constr. Approx., 35:2 (2012), 181–200  crossref  mathscinet  zmath  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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