Asymptotically optimal unsaturated lattice cubature formulae with bounded boundary layer

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
$$ \gamma\lt \frac12, \qquad c_k(h)=1, \quad\text{if \ }\rho(hk,\mathbb R^n\setminus\Omega)\geqslant Ch^{\gamma}. $$
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.