Abstract
We study the problem of the optimization of approximate integration on the class of functions defined on the parallelepiped Πd=[0,a1]×⋅⋅⋅×[0,ad], a1,…,ad>0, having a given majorant for the modulus of continuity (relative to the l1-metric in ℝd). An optimal cubature formula, which uses as information integrals of f along intersections of Πd with n arbitrary (d−1)-dimensional hyperplanes in ℝd (d>1) is obtained. We also find an asymptotically optimal sequence of cubature formulas, whose information functionals are integrals of f along intersections of Πd with shifts of (d−2)-dimensional coordinate subspaces of ℝd (d>2).
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