Optimal Cubature Formulas Research Articles

CUBATURE FORMULAS FOR CLASSES OF FUNCTIONS WITH BOUNDED MIXED DIFFERENCE

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). Bibliography: 9 titles.

Optimal cubature formulae for computing many-dimensional integrals of functions in the class [formula omitted

Order optimal cubature formulae are constructed for evaluating integrals of functions in class Q r, τ(Ω, 1) , where Ω=[−1, 1] 1 , ι⩾2. Asymptotically optimal quadrature formulae are constructed for evaluating integrals of functions in Q r, τ(Ω, 1) where Ω=[−1, 1].

Sets of convergence to zero of lacunary trigonometric series

Nauka, Moscow (1969). 6. L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York (1974). 7. Hua Loo Keng and Wang Yuan, Applications of Number Theory to Numerical Analysis, Springer-Verlag, Berlin (1981). 8. N. S. Bakhvalov, "On approximate computation of multiple integrals," Vestn. Mosk. Gos. Univ., Ser. Mat., Fiz., Astron., Khim., No. 4, 3-18 (1959). 9. I. F. Sharygin, "Lower bounds for the errors of quadrature formulas on function classes," Zh. Vychisl. Mat. Mat. Fiz., ~, No. 2, 370-376 (1963). i0. N. S. Bakhvalov, "Lower estimates for asymptotic characteristics of classes of functions with a dominating mixed derivative," Mat. Zametki, 12, No. 6, 655-664 (1972). ii. N. S. Bakhvalov, "On the optimal bounds of convergence of quadrature processes and of integration methods of Monte Carlo type on function classes," Zh. Vychisl. Mat. Mat. Fiz., ~, No. 4, Supplement, 5-63 (1964). 12. V. A. Bykovskii, "On the correct order of the error of optimal cubature formulas in spaces with dominating derivative and on square deviations of nets," Preprint, Akad. Nauk SSSR, Far Eastern Science Center (1985), pp. 3-31. 13. K. K. Frolov, "Upper bounds for the errors of quadrature formulas on function classes," Dokl. Akad. Nauk SSSR, 231, No. 4, 818-821 (1976). 14. V. N. Temlyakov, "On the reconstruction of periodic functions of several variables from the values at the nodes of number-theoretic nets," Analysis Mathematica, Akad. Kiado, Budapest, 12, No. 4, 287-305 (1986). 15. V. N. Temlyakov, "On the approximation by elements of a finite-dimensional subspace of functions from various classes of Sobolev or Nikol'skii," Mat. Zametki, 43, No. 6, 770-785 (1988).

On the approximate calculation of double integrals

Cubature formulas are obtained which are optimal or asymptotically optimal on given sets of functions. These formulas consist of line integrals which may be evaluated by optimal or asymptotically optimal quadrature formulas. The advantage of these formulas over the optimal and asymptotically optimal cubature formulas with rectangular-lattices of knots is shown.

On the Evaluation of Double Integrals

A cubature formula consisting of line integrals which is optimal on a set of functions satisfying given boundary conditions is obtained. The line integrals of this formula may be evaluated by optimal quadrature formulas. The advantage of this formula over the optimal cubature formula with a rectangular lattice of knots is shown. This approach to optimal cubatures was stimulated by the idea of blending [1], [2].

On the evaluation of double integrals

A cubature formula consisting of line integrals which is optimal on a set of functions satisfying given boundary conditions is obtained. The line integrals of this formula may be evaluated by optimal quadrature formulas. The advantage of this formula over the optimal cubature formula with a rectangular lattice of knots is shown. This approach to optimal cubatures was stimulated by the idea of blending [1], [2].