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^mspaces, 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:

Sbornik: Mathematics
 Pub Date:
 July 2013
 DOI:
 10.1070/SM2013v204n07ABEH004328
 Bibcode:
 2013SbMat.204.1003R