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Sets of lattice cubature rules, which are precise for trigonometric polynomials of three variables and are the best ones with respect to the number of nodes. (Russian, English) Zbl 1079.41030
Zh. Vychisl. Mat. Mat. Fiz. 45, No. 2, 212-223 (2005); translation in Comput. Math. Math. Phys. 45, No. 2, 202-212 (2005).
There are considered the lattice cubature formulas with the lattice of nodes \(\Lambda_k = M_k^{\perp}\), where the lattice is generated by a matrix \(kB + C\). Here \(B, C\) are integer numbers quadratic matrices of \(n\)-th order, which do not depend on \(k\) and \(\det(B) \neq 0 \). At \(n = 3\) for any integer number \(r, (-4 \leq r \leq 1)\) there is indicated a run \(S^{(\min)}\) with a trigonometric \((6k + r)\)-property, which has an asymptotically small number of nodes.

MSC:
41A55 Approximate quadratures
65D32 Numerical quadrature and cubature formulas
42A10 Trigonometric approximation
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