Geodesic
Asymptotically optimal unsaturated lattice cubature formulae with bounded boundary layer
Sbornik. Mathematics, Tome 204 (2013) no. 7, pp. 1003-1027 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper describes a new algorithm for constructing lattice cubature formulae with bounded boundary layer $$ \int_{\Omega}f(x)\, dx\approx h^n\sum_{\substack{k\in\mathbb{Z}^n \\ \rho(hk,\Omega)\leqslant Ch^{\gamma}}} c_k(h) f(hk), $$ where $$ \gamma<\frac12, \qquad c_k(h)=1, \quad\text{if \ }\rho(hk,\mathbb R^n\setminus\Omega)\geqslant Ch^{\gamma}. $$ These formulae are unsaturated (in the sense of Babenko) both with respect to the order and in regard to the property of asymptotic optimality on $W_2^m$-spaces, $m\in(n/2,\infty)$. Most of the results obtained apply also to $W_2^\mu(\mathbb{R}^n)$-spaces with a hypoelliptic multiplier of smoothness $\mu$. Bibliography: 6 titles.
Keywords: optimization, unsaturated algorithm.
Mots-clés : cubature formulae 
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     author = {M. D. Ramazanov},
     title = {Asymptotically optimal unsaturated lattice cubature formulae with bounded boundary layer},
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     url = {http://geodesic.mathdoc.fr/item/SM_2013_204_7_a3/}
}
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M. D. Ramazanov. Asymptotically optimal unsaturated lattice cubature formulae with bounded boundary layer. Sbornik. Mathematics, Tome 204 (2013) no. 7, pp. 1003-1027. http://geodesic.mathdoc.fr/item/SM_2013_204_7_a3/

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