Sets of lattice cubature rules that are optimal in terms of the number of nodes and exact on trigonometric polynomials in three variables

N. N. Osipov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 2, pp. 212-223

Résumé

Sets of lattice cubature rules with the lattice of nodes $\lambda_k=M_k^\perp$, where the lattice $M_k$ is generated by the matrix $kB+C$ ($B$ and $C$ are integer square matrices of order $n$ independent of $k$ and $\det(B)\ne 0$) are considered. At $n=3$, for each integer $r$ ($-4\le r\le 1$), the set $S^{(\min)}$ with the trigonometric $(6k+r)$ property and the asymptotically minimal number of nodes $N^{(\min)}(k)$ is found. This means that, for any set $S^{(\min)}$ with the trigonometric $(6k+r)$ property and the number of nodes $N(k)$, the inequality $N(k)\ge N^{(min)}(k)$ holds true if $k$ is sufficiently large. Certain properties of the optimal sets $S^{(min)}$ and the nearest (in terms of the number of nodes) sets $S^{(\min+)}$ are investigated.