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Construction of sequences of the rank 1 lattice cubature formulas that are exact on trigonometric polynomials. (Russian, English) Zbl 1086.65508
Zh. Vychisl. Mat. Mat. Fiz. 42, No. 11, 1627-1635 (2002); translation in Comput. Math. Math. Phys. 42, No. 11, 1563-1571 (2002).
A simple method for constructing sequences of rank 1 lattice cubature formulas that are exact for trigonometric polynomials in $$n$$ variables ($$n\geq 2$$) is proposed. It is shown how to construct an infinite series of rank 1 lattice cubature formulas having the trigonometric $$d(k)$$-property, where $$d(k)=(d_{0}+1)k- D$$ and $$D$$ and $$D$$ is independent of $$k$$, on the basis of a single lattice cubature formula of an arbitrary rank having the trigonometric $$d_0$$-property. The cubature formulas in the resulting series are asymptotically equivalent to the original cubature formula in terms of efficiency (which characterizes the quality of the cubature formulas under consideration). In the $$n$$-dimensional case, a sequence with an efficiency of $$4^{n-1}/n$$ is constructed by this method, which significantly improves earlier results for $$n \geq 5$$. For $$n=3$$, a sequence with the maximum possible efficiency $$108/19$$ is presented. For $$n=4$$ and $$5$$, a technique for constructing sequences with efficiencies higher than $$4^{n-1}/n$$ is shown.