Geodesic
Estimation of the remainder of a cubature formula on a Chebyshev grid for two-variable functions
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 7, pp. 1185-1191 
V. A. Abilov; M. K. Kerimov. Estimation of the remainder of a cubature formula on a Chebyshev grid for two-variable functions. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 7, pp. 1185-1191. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_7_a1/
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     author = {V. A. Abilov and M. K. Kerimov},
     title = {Estimation of the remainder of a cubature formula on a {Chebyshev} grid for two-variable functions},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1185--1191},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\lambda$.

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