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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 $$\mathbb R^{ d } (d>2)$$.

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