On an Optimal Quadrature Formula for Classes of Functions Given by Modulus of Continuity
https://doi.org/10.18255/1818-1015-2014-3-91-105
Abstract
The problem of minimizing the error of a cubature formula on the classes of functions given by modulus of continuity for cubature formulas with fixed nodes on the boundary of gird rectangular localization domain of nodes is considered. We give the exact solution of this problem on the wide classes of functions of two variables. It was previously shown by N.P. Korneychuk that if the boundary nodes of a rectangular lattice Qk,i = { xk-1 ≤ x ≤ xk , yi-1 ≤ y ≤ yi} are not included in the number of nodes cubature formula
Z Z (Q) f(x, y)dxdy = Xm k=1 Xn i=1 pkif(xk, yi) + Rmn(f), (1)
the formula of average rectangles is the best for classes of functions ω1,ω2 (Q), Hω1p1 (Q) and Hω1p2(Q) among all quadrature formulas of the form (1). It is proved that if into the number of nodes in the formula (1) all boundary nodes (such formulas are called Markov-type) are added, then for these classes of functions the best formula is trapezoids. The exact errors for all classes of functions are calculated.
About the Author
M. Sh. Shabozov
A. Juraev Institute of mathematics, Academy of Sciences of the Republic Tajikistan
Tajikistan
доктор физ.-мат. наук, академик АН Республики Таджикистан, профессор, A˘ıni Street, 299/4, Dushanbe city, 734063, Tajikistan
Tajikistan
доктор физ.-мат. наук, академик АН Республики Таджикистан, профессор, A˘ıni Street, 299/4, Dushanbe city, 734063, Tajikistan
References
1. Никольский С.М. Квадратурные формулы. М.: Наука, 1988. (English transl.: Nikol’skiĭ S.M. Quadrature formulas. Moskva: Nauka, 1988.)
2. Корнейчук Н.П. Наилучшие кубатурные формулы для классов функций многих переменных // Математические заметки. 1968. Т. 3, №5. С. 565–576. (English transl.: Korneĭchuk N.P. Best cubature formulas for certain classes of functions of several variables // Matematicheskie Zametki. 1968. V. 3, №5. P. 565–576.)
3. Корнейчук Н.П. Точные константы в теории приближения. М.: Наука, 1987. (Korneĭchuk N.P. Exact Constants in Approximation Theory. Moskva: Nauka, 1987; English transl. in Encyclopedia Math. Appl., V. 38, Cambridge Univ. Press, Cambridge, 1991.)
Review
For citations:
Shabozov M.Sh. On an Optimal Quadrature Formula for Classes of Functions Given by Modulus of Continuity. Modeling and Analysis of Information Systems. 2014;21(3):91-105. (In Russ.) https://doi.org/10.18255/1818-1015-2014-3-91-105
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