Abstract:
Cubature formulas are considered for classes of functions with bounded mixed difference. The correct order is found for the error of optimal cubature formulas that use the values of the functions at particular points. It turns out that, for the classes indicated above, cubature formulas constructed with the help of number-theoretic methods are optimal (in the sense of order).
Citation:
V. V. Dubinin, “Cubature formulas for classes of functions with bounded mixed difference”, Russian Acad. Sci. Sb. Math., 76:2 (1993), 283–292
\Bibitem{Dub92}
\by V.~V.~Dubinin
\paper Cubature formulas for classes of functions with bounded mixed difference
\jour Russian Acad. Sci. Sb. Math.
\yr 1993
\vol 76
\issue 2
\pages 283--292
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\crossref{https://doi.org/10.1070/SM1993v076n02ABEH003413}
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\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?1993SbMat..76..283D}
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Linking options:
https://www.mathnet.ru/eng/sm1054
https://doi.org/10.1070/SM1993v076n02ABEH003413
https://www.mathnet.ru/eng/sm/v183/i7/p23
This publication is cited in the following 9 articles:
D. B. Bazarkhanov, “Optimal Cubature Formulas on Classes of Periodic Functions in Several Variables”, Proc. Steklov Inst. Math., 312 (2021), 16–36
Christopher Kacwin, Jens Oettershagen, Mario Ullrich, Tino Ullrich, “Numerical Performance of Optimized Frolov Lattices in Tensor Product Reproducing Kernel Sobolev Spaces”, Found Comput Math, 21:3 (2021), 849
Van Kien Nguyen, Ullrich M., Ullrich T., “Change of Variable in Spaces of Mixed Smoothness and Numerical Integration of Multivariate Functions on the Unit Cube”, Constr. Approx., 46:1 (2017), 69–108
Ullrich M., Ullrich T., “The Role of Frolov's Cubature Formula for Functions with Bounded Mixed Derivative”, SIAM J. Numer. Anal., 54:2 (2016), 969–993
Aicke Hinrichs, Lev Markhasin, Jens Oettershagen, Tino Ullrich, “Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions”, Numer. Math., 134:1 (2016), 163
Temlyakov V., “Cubature Formulas, Discrepancy, and Nonlinear Approximation”, J. Complex., 19:3 (2003), 352–391
V. V. Dubinin, “Cubature formulae for Besov classes”, Izv. Math., 61:2 (1997), 259–283
N. Temirgaliev, “Efficiency of numerical integration algorithms related to divisor theory in cyclotomic fields”, Math. Notes, 61:2 (1997), 242–245
Heping Wang, “Quadrature formulas for classes of functions with bounded mixed derivative or difference”, Sci China Ser A, 40:5 (1997), 449
Citation in format AMSBIB
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