Estimation of the remainder of a cubature formula on a Chebyshev grid for two-variable functions |
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Authors: | V A Abilov M K Kerimov |
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Institution: | 1. Dagestan State University, ul. Gadzhieva 43a, Makhachkala, 367015, Russia 2. Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia
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Abstract: | For a cubature formula of the form $$\int\limits_0^{2\pi } {\int\limits_0^{2\pi } {f(x,y)dxdy = \frac{{4\pi ^2 }} {{mn}}\sum\limits_{i = 0}^{n - 1} {\sum\limits_{j = 0}^{m - 1} {f\left( {\frac{{2\pi i}} {n},\frac{{2\pi j}} {m}} \right) + R_{n,m} (f)} } } }$$ on a Chebyshev grid, the remainder R n,m (f) is proved to satisfy the sharp estimate $$\mathop {\sup }\limits_{f \in H\left( {r_1 ,r_2 } \right)} \left| {R_{n,m} (f)} \right| = O\left( {n^{ - r_1 + 1} + m^{ - r_1 + 1} } \right)$$ in some class of functions H(r 1, r 2) defined by a generalized shift operator. Here, r 1, r 2 > 1; ???1 ?? n/m ?? ?? with ?? > 0; and the constant in the O-term depends only on ??. |
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