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Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2012, Volume 52, Number 8, Pages 1373–1377
(Mi zvmmf9704)
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This article is cited in 1 scientific paper (total in 1 paper)
Estimation of the remainder of a cubature formula on a Chebyshev grid
V. A. Abilov, M. K. Kerimov Daghestan State University
Abstract:
Let $C(Q)$ denote the space of continuous functions $f(x,y)$ in the square $Q=[-1,1]\times[-1,1]$ with the norm
\begin{equation}
\| f\|=\max(|(f(x,y)|),
\quad
(x,y)\in Q
\end{equation}
On a Chebyshev grid, a cubature formula of the form
\begin{eqnarray}
&\int_{-1}^1\int_{-1}^1\frac{1}{\sqrt{(1-x^2)(1-y^2)}}f(x,y)dxdy=
\\
&\frac{\pi^2}{mn}\sum_{i=1}^n\sum_{j=1}^mf\big (\cos\frac{2i-1}{2n}\pi,cos\frac{2j-1}{2m}\pi\big )+R_{m,n}(f)
\end{eqnarray}
is considered in some class $H(r_1,r_2)$ of functions $f\in C(Q)$, defined by a generalized shift operator. The remainder $R_{m,n}(f)$ is proved to satisfy the estimate:
$$
\sup_{f\in H(r_1,r_2)}| R_{m,n}(f) |=O(n^{-r_1+1}+m^{-r_2+1})
$$
where $r_1,r_2>1,\lambda^{-1}\leq n/m\leq\lambda,\lambda>0$; and the constant in $O(1)$, depends on $\lambda$. Библ. 4.
Ключевые слова: кубатурная формула, чебышевская сетка, оценка остаточного члена.
Received: 21.03.2012
Citation:
V. A. Abilov, M. K. Kerimov, “Estimation of the remainder of a cubature formula on a Chebyshev grid”, Zh. Vychisl. Mat. Mat. Fiz., 52:8 (2012), 1373–1377; Comput. Math. Math. Phys., 52:8 (2012), 1089–1093
Linking options:
https://www.mathnet.ru/eng/zvmmf9704 https://www.mathnet.ru/eng/zvmmf/v52/i8/p1373
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