Construction of Optimal Cubature Formulas Related to Computer Tomography

We study the problem of the optimization of approximate integration on the class of functions defined on the parallelepiped Π d =[0, a 1 ]×⋅⋅⋅×[0, a d ], a 1 ,…, a d >0, having a given majorant for the modulus of continuity (relative to the l 1 -metric in ℝ d ). An optimal cubature formula, which...

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Bibliographic Details
Published inConstructive approximation Vol. 33; no. 3; pp. 313 - 330
Main Authors Babenko, V. F., Borodachov, S. V., Skorokhodov, D. S.
Format Journal Article
LanguageEnglish
Published New York Springer-Verlag 01.06.2011
Subjects
Analysis
Mathematics
Mathematics and Statistics
Numerical Analysis
41A55
65D30
Optimal cubature formula
41A63
65D32
Radon transform
Modulus of continuity
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Summary:We study the problem of the optimization of approximate integration on the class of functions defined on the parallelepiped Π d =[0, a 1 ]×⋅⋅⋅×[0, a d ], a 1 ,…, a d >0, having a given majorant for the modulus of continuity (relative to the l 1 -metric in ℝ d ). An optimal cubature formula, which uses as information integrals of f along intersections of Π d with n arbitrary ( d −1)-dimensional hyperplanes in 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 Π d with shifts of ( d −2)-dimensional coordinate subspaces of ℝ d ( d >2).
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-010-9095-6