Optimal cubature formulas arising from computer tomography

The problem studied in this talk concerns optimal formula of approximate integration along a d-dimensional parallelepiped D, which uses as information about the function its mean values along intersections of D and n arbitrary (d-1)-dimensional hyperplanes. We find a cubature formula with the smallest supremum of the absolute error over the class of functions continuous on $D$, which have a given majorant for their moduli of continuity with respect to the "sum" norm. We prove that the node hyperplanes of the optimal cubature formula are perpendicular to the shortest edge of D and are equally spaced. On the class of functions, which are continuous on a d-dimensional cube C and have a given majorant for their moduli of continuity with respect to the "max" norm, we find an optimal formula, which recovers integral along C from integrals along intersections of C and n shifts of k-dimensional coordinate subspaces, 0. Here n is assumed to be a (d-k)-th power of an integer.