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

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

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
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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
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
3. Kroo A., “Rate of Approximation of Regular Convex Bodies by Convex Algebraic Level Surfaces”, Constr. Approx., 35:2 (2012), 181–200
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