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Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2015, Volume 15, Issue 2, Pages 151–160
DOI: https://doi.org/10.18500/1816-9791-2015-15-2-151-160
(Mi isu576)
 

This article is cited in 5 scientific papers (total in 5 papers)

Mathematics

Isoperimetry coefficient for simplex in the problem of approximation of derivatives

V. A. Klyachin, D. V. Shurkaeva

Saratov State University, 83, Astrakhanskaya st., 410012, Saratov, Russia
Full-text PDF (219 kB) Citations (5)
References:
Abstract: We introduce the isoperimetry coefficient $\sigma(G)= {|\partial G|^{n/(n-1)}}/{|G|} $ of region $G\subset \mathbb R^n$. In terms of this the error $\delta_\Delta(f)$ estimates for the gradient of the piecewise linear interpolation of functions of class $C^1(G)$, $C^2(G)$, $C^{1,\alpha}(G)$, $0<\alpha<1$, are obtained. The problem of obtaining such estimates is nontrivial, especially in the multidimensional case. Here it should be noted that in the two-dimensional case, for functions of class $C^2(G)$, the convergence of the derivatives is provided by the classical Delaunay condition. In the multidimensional case, as shown by the examples, such conditions are not sufficient. Nevertheless, the article shows how to apply these estimates to the Delaunay triangulation of multidimensional discrete $ \varepsilon $-nets. The results obtained give sufficient conditions for convergence of the derivatives on the Delaunay triangulation of discrete $ \varepsilon $-nets with $ \varepsilon \to 0 $. In addition, the ratio of the distortion factor is found for isoperimetry coefficient under the quasi-isometric transformation.
Key words: isoperimetry coefficient, simplex, piecewise linear interpolation.
Bibliographic databases:
Document Type: Article
UDC: 514.174.3+519.65
Language: Russian
Citation: V. A. Klyachin, D. V. Shurkaeva, “Isoperimetry coefficient for simplex in the problem of approximation of derivatives”, Izv. Saratov Univ. Math. Mech. Inform., 15:2 (2015), 151–160
Citation in format AMSBIB
\Bibitem{KlyShu15}
\by V.~A.~Klyachin, D.~V.~Shurkaeva
\paper Isoperimetry coefficient for simplex in the problem of approximation of~derivatives
\jour Izv. Saratov Univ. Math. Mech. Inform.
\yr 2015
\vol 15
\issue 2
\pages 151--160
\mathnet{http://mi.mathnet.ru/isu576}
\crossref{https://doi.org/10.18500/1816-9791-2015-15-2-151-160}
\elib{https://elibrary.ru/item.asp?id=23647131}
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  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика
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