Cubature formulas of S. L. Sobolev: evolution of the theory and applications

The paper contains the description of the theory of approximate calculation of integrals over arbitrary multi-dimensional domains. This research branch is developed in several research centers in Russia and, in particular, in the Ufa Mathematical Institute of the Russian Academy of Sciences. We consider the best approximations of linear functionals on a certain semi-Banach space $B$ by linear combinations of the Dirac functions with supports in the nodes of a certain lattice:
$$ (l_N,f)\equiv\int\limits_\Omega f(x)\,dx-\sum_{k\in\mathbb{Z}^n,\atop H_N k\in\Omega}c_k f(H_N k), $$
where $H_N$ is an $n\times n$ matrix, such that $\det H_N\ne 0$ and $\det H_N\to 0$ as $N\to\infty$ and $f\colon\mathbb{R}^n\to\mathbb{C}$, $f\in B\subset C(\mathbb{R}^n)$.
This setting of the problem was given by academician Sergei L'vovich Sobolev in the middle of the last century.