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Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2012, Volume 52, Number 7, Pages 1185–1191
(Mi zvmmf9597)
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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
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
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
Linking options:
https://www.mathnet.ru/eng/zvmmf9597 https://www.mathnet.ru/eng/zvmmf/v52/i7/p1185
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Abstract page: | 281 | Full-text PDF : | 83 | References: | 47 | First page: | 10 |
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