General algorithm for the numerical integration of functions of several variables

An algorithm is proposed for the numerical integration of an arbitrary function representable as a sum of an absolutely converging multiple trigonometric Fourier series. The resulting quadrature formulas have identical weights, and the nodes form a Korobov grid that is completely defined by two positive integers, of which one is the number of nodes. In the case of classes of functions with dominant mixed smoothness, it is shown that the algorithm is almost optimal in the sense that the construction of a grid of $N$ nodes requires far fewer elementary arithmetic operations than $N\ln\ln N$. Solutions of related problems are also given.