ABSTRACT. Cubature formulas for calculating integrals over the hyperoctahedron that are invariant under the group of all of its orthogonal transformations are obtained. Two of them are exact for all polynomials of degree no greater than seven and one is exact for all polynomials of degree no greater than five.KEY WORDS: cubature formulas, polynomials, invariance under symmetries, hyperoctahedron. (The hyperoctahedron On is the polyhedron in 1~ with vertices at the 2n pointsThe group of all orthogonal transformations of On into itself will be denoted by OriG. It is known [1] that the order of this group equals n!2 ~.Cubature formulas for the hyperoctahedron O~ invariant under transformations from the group OnG and exact for all polynomials of degree less than or equal to three and five are given in [2].In this paper, we use Sobolev's theorem [3] for the hyperoctahedron O~ to construct a cubature formula exact for all polynomials of degree less than or equal to five and two cubature formulas exact for polynomials of degree less than or equal to seven. These formulas are invariant under the group OriG.
517.518.87 In the paper, cubature formulas for calculation of an integral along the hypercube are constructed. These formulas are invariant under the group of all transformations of the hyperoctahedron and precise for all polynomials of degree less than or equal to 7. Bibliography: 2 titles Consider the cubature formula N K. j=l (i) where x = (Xl, x2,... , Xn) and K,, is the hypercube K,, := {x E •n{ -l <_ zi <_ l, i = l,2,... ,n}.
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