Optimal cubature formulas for tensor products of certain classes of functions

We study the problem of constructing an optimal formula of approximate integration along a dd-dimensional parallelepiped. Our construction utilizes mean values along intersections of the integration domain with nn hyperplanes of dimension (d−1)(d−1), each of which is perpendicular to some coordinate axis. We find an optimal cubature formula of this type for two classes of functions. The first class controls the moduli of continuity with respect to all variables, whereas the second class is the intersection of certain periodic multivariate Sobolev classes. We prove that all node hyperplanes of the optimal formula in each case are perpendicular to a certain coordinate axis and are equally spaced and the weights are equal. For specific moduli of continuity and for sufficiently large nn, the formula remains optimal for the first class among cubature formulas with arbitrary positions of hyperplanes.