Asymptotically optimal unsaturated lattice cubature formulae with bounded boundary layer
Abstract
This paper describes a new algorithm for constructing lattice cubature formulae with bounded boundary layer
\displaystyle \int_{\Omega}f(x)\, dx\approx h^n\sum_{\substack{k\in{Z}^n \\ \rho(hk,\Omega)\le Ch^{\gamma}}} c_k(h) f(hk), where \displaystyle \gamma<\frac12, \qquad c_k(h)=1, if \displaystyle \rho(hk, R^n\setminus\Omega)\ge 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({R}^n)-spaces with a hypoelliptic multiplier of smoothness \mu. Bibliography: 6 titles.- Publication:
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Sbornik: Mathematics
- Pub Date:
- July 2013
- DOI:
- 10.1070/SM2013v204n07ABEH004328
- Bibcode:
- 2013SbMat.204.1003R