Approximation of functions of many variables from the generalization Nikol'skii-Besov type classes in the uniform and integral metrics

We obtain the exact order estimates of the approximation of the functions of many variables from the generalized Nikol’skii-Besov classes $B_{p,\theta }^{\mathrm{\Omega }}({\mathbb{R}}^d)$ by sums of de la Vallee Poussin type in the metrics space $L_{\mathrm{\infty }}({\mathbb{R}}^d)$ and $L_1({\mathbb{R}}^d)$. These classes of functions for some given $\mathrm{\Omega }$ coincide with the well-known classical Nikol’skii-Besov isotropic classes.