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Construction of optimal cubature formulas related to computer tomography. (English) Zbl 1231.65054
Summary: We study the problem of the optimization of approximate integration on the class of functions defined on the parallelepiped \(\Pi _{ d }=[0,a _{1}]\times \cdot \cdot \cdot \times [0,a _{ d }], a _{1},\cdots ,a _{ d }\)>0, having a given majorant for the modulus of continuity (relative to the \(l _{1}\)-metric in \(\mathbb R^{ d }\)). An optimal cubature formula, which uses as information integrals of \(f\) along intersections of \(\Pi _{ d }\) with \(n\) arbitrary \((d - 1)\)-dimensional hyperplanes in \(\mathbb R^{ d } (d>1)\) is obtained. We also find an asymptotically optimal sequence of cubature formulas, whose information functionals are integrals of \(f\) along intersections of \(\Pi _{ d }\) with shifts of \((d - 2)\)-dimensional coordinate subspaces of \(\mathbb R^{ d } (d>2)\).

65D30 Numerical integration
65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
41A63 Multidimensional problems
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