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Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2012, Volume 52, Number 7, Pages 1185–1191 (Mi zvmmf9597)  

This article is cited in 4 scientific papers (total in 4 papers)

Estimation of the remainder of a cubature formula on a Chebyshev grid for two-variable functions

V. A. Abilova, M. K. Kerimovb

a Dagestan State University, ul. Gadzhieva 43a, Makhachkala, 367015 Russia
b Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333 Russia
Full-text PDF (195 kB) Citations (4)
References:
Abstract: For a cubature formula of the form
$$ \int_0^{2\pi}\int_0^{2\pi}f(x,y)\,dx\,dy=\frac{4\pi^2}{mn}\sum_{i=0}^{n-1}\sum_{j=0}^{m-1} f\biggl(\frac{2\pi i}{n},\frac{2\pi j}{m}\biggr)+R_{n,m}(f). $$
on a Chebyshev grid, the remainder $R_{n,m}(f)$ is proved to satisfy the sharp estimate
$$ \sup_{f\in H(r_1,r_2)}|S_{n,m}(f)|=O(n^{-r_1+1}+m^{-r_1+1}) $$
in some class of functions $H(r_1,r_2)$ defined by a generalized shift operator. Here, $r_1,r_2>1$; $\lambda^{-1}\le n/m\le\lambda$ with $\lambda>0$ and the constant in the $O$-term depends only on $\lambda$.
Key words: cubature formula on a Chebyshev grid, remainder, remainder estimate, generalized shift operator.
Received: 18.01.2012
English version:
Computational Mathematics and Mathematical Physics, 2012, Volume 52, Issue 7, Pages 985–991
DOI: https://doi.org/10.1134/S0965542512070020
Bibliographic databases:
Document Type: Article
UDC: 519.644.7
Language: Russian
Citation: V. A. Abilov, M. K. Kerimov, “Estimation of the remainder of a cubature formula on a Chebyshev grid for two-variable functions”, Zh. Vychisl. Mat. Mat. Fiz., 52:7 (2012), 1185–1191; Comput. Math. Math. Phys., 52:7 (2012), 985–991
Citation in format AMSBIB
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  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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