Український математичний конгрес  2009
Sergiy Borodachov (Towson University, Towson, USA) Optimal cubature formulas arising from computer tomography The problem studied in this talk concerns optimal formula of
approximate integration along a ddimensional parallelepiped D,
which uses as information about the function its mean values along
intersections of D and n arbitrary (d1)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 ddimensional
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 kdimensional coordinate subspaces, 0
