Abstract
Optimal cubature formulas are constructed for calculations of multidimensional integrals in weighted Sobolev spaces. We consider some classes of functions defined in the cube Ω = [-1, 1]l, l = 1, 2,..., and having bounded partial derivatives up to the order r in Ω and the derivatives of jth order (r < j ≤ s) whose modulus tends to infinity as power functions of the form (d(x, Г))-(j-r), where x ∈ Ω Г, x = (x 1,..., x l ), Г = ∂Ω, and d(x, Г) is the distance from x to Г.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Kolmogoroff A., “Zur Grossenordnung des Restgleides Fourierscher Reihen differenzierbarer Funktionen,” Ann. Math., Bd 36, 521–526 (1935).
Nikol’skiĭ S. M., “Concerning estimation for approximate quadrature formulas,” Uspekhi Mat. Nauk, 5, No. 2, 165–177 (1950).
Bakhvalov N. S., “Properties of optimal methods for the solution of problems of mathematical physics,” USSR Comp. Math. Math. Phys., 10, No. 3, 1–19 (1970).
Babenko K. I. (ed.), Theoretical Foundations and Construction of Numerical Algorithms of Problems of Mathematical Physics [in Russian], Nauka, Moscow (1979).
Traub J. F. and Wozniakowski H., A General Theory of Optimal Algorithms, Academic Press, New York (1980).
Nikol’skiĭ S. M., Quadrature Formulas [in Russian], Nauka, Moscow (1979).
Bakhvalov N. S., “On optimal estimates for convergence of quadrature processes and integration methods of Monte Carlo type on function classes,” in: Numerical Methods for Solving Differential and Integral Equations and Cubature Formulas [in Russian], Nauka, Moscow, 1964, pp. 5–63.
Sobolev S. L., Introduction to the Theory of Cubature Formulas [in Russian], Nauka, Moscow (1974).
Sobolev S. L. and Vaskevich V. L., The Theory of Cubature Formulas, Kluwer Academic Publishers, Dordrecht (1997).
Polovinkin V. I., “Questions in the theory of weight of cubature formulas,” Sib. Math. J., 12, No. 1, 128–142 (1971).
Babenko V. F., “Faithful asymptotics of remainders optimal for some classes of functions with cubic weight formulas,” Math. Notes, 20, No. 4, 887–890 (1976).
Boikov I. V., “Optimal cubature formulae for computing many-dimensional integrals of functions in the class,” USSR Comp. Math. Math. Phys., 30, No. 4, 110–117 (1990).
Babenko K. I., “Some problems in approximation theory and numerical analysis,” Russian Math. Surveys, 40, No. 1, 1–30 (1985).
Boikov I. V., The Optimal Methods of Approximation of the Functions and Computing the Integrals [in Russian], Penza State University, Penza (2007).
Boikov I. V., Approximation Methods for Computing Singular and Hypersingular Integrals. Part 2: Hypersingular Integrals [in Russian], Penza State University, Penza (2009).
Nikol’skiĭ S. M., A Course of Mathematical Analysis. Vol. 1, Nauka, Moscow (1975).
Author information
Authors and Affiliations
Corresponding author
Additional information
Penza. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 3, pp. 543–561, May–June, 2016; DOI: 10.17377/smzh.2016.57.305.
Rights and permissions
About this article
Cite this article
Boikov, I.V. Optimal cubature formulas for calculation of multidimensional integrals in weighted Sobolev spaces. Sib Math J 57, 425–441 (2016). https://doi.org/10.1134/S0037446616030058
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446616030058