Extremal cubature formulas for anisotropic classes

Let $E^{(\alpha; s)}$ be a class of periodical functions
$$f(x_1, \dots, x_s)=\sum_{(m_1, \dots, m_s)\in \mathbb{Z}^s} c(m_1, \dots, m_s)\exp\left(2\pi i(m_1 x_1+\dots+ m_s x_s)\right)$$
with $\left|c(m_1, \dots, m_s)\right|\leq \prod_{j=1} \left(\text{max} (1, |m_j|)\right)^{-\alpha},$ and $1< \alpha < \infty$. In this work for all natural numbers $1< N < \infty$ we prove best possible estimation
$$R_N\left(E^{(\alpha; s)}\right)\ll_{\alpha, s} \frac{\left(\log N\right)^{s-1}}{N^\alpha}$$
for the error of the best cubature formula on the class $E^{(\alpha; s)}$ with $N$ nodes and weights. Similar results are proved for other classes of functions.