New algorithm of asymptotically optimal lattice cubature formulae

Lattice cubature formulae are used in approximate computation of integrals for smooth functions of several variables $\int_\Omega f(x)\,dx$ by means of linear combinations $h^n\sum\limits_{\substack{k\in\mathbb Z^n,\\hk\in\Omega}}c_kf(hk)$. Asymptotically optimal formula in the $W_2^m$-space is defined by
\begin{multline*} \sup_{f\in W_2^m(\Omega)}\Bigl|\int_\Omega f(x)\,dx-h^n\sum_{hk\in\Omega}c_k^{as}f(hk)\Bigr|/\\ \inf_{\{c_k\}}\sup_{f\in W_2^m(\Omega)}\Bigl|\int_\Omega f(x)\,dx-h^n\sum_{hk\in\Omega}c_kf(hk)\Bigr|=1. \end{multline*}

K. I. Babenko proposed the concept of unsaturated computational algorithms [7] – for preserving the optimal orders of convergence for all spaces of functions being parameters of the problem.
A new algorithm for constructing the lattice cubature formulas, unsaturated not only by the order, but also by the property of asymptotic optimality in the $W_2^m$-spaces, $m\in(n/2,\infty)$ is described in the paper.