An invariant cubature formula of degree nine for a hypercube

In this work, cubature formulas for computation of integrals over the hypercube in R ⁿ

$C_n = \{ x = (x_1 ,x_2 ,...,x_n ) \in R^n | - 1 \leqslant x_i \leqslant 1,i = 1,2,...,n\} $

are constructed using Sobolev?s theorem. These formulas are precise for all polynomials of degree at most nine and are invariant under the group of all orthogonal transformations of the hyperoctahedron

$G_n = \left\{ {x = (x_1 ,x_2 ,...,x_n ) \in R^n |\sum\limits_{i = 1}^n {|x_i |} \leqslant 1} \right\}$

onto itself.

Section 1 contains introduction into the subject and a review of known results. In Sections 2 and 3, we determine parameters of the cubature formula for n = 4 and n = 3, respectively. Numerical results (the nodes and coefficients of the cubature formulas) are presented in Section 4.