The best convergence of multiple trigonometric series

We consider the best convergence of multiple trigonometric series. We indicate essential distinction of the behavior (in this sense) of multiple series from that of simple ones. In particular, the well-known result obtained by S. N. Bernshtein on the best convergence of a series with an odd ratio of frequencies does not hold for a multiple series in the case of the approximation by polynomials with harmonics from rectangles (in the sense of Pringsheim), but it is true for “angular” approximations.