Extremal cubature formulas for anisotropic classes

V. A. Bykovskii

Dalʹnevostočnyj matematičeskij žurnal, Tome 19 (2019) no. 1, pp. 10-19

Résumé

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.