Информационная система "Конференции"

Abstracts

Approximation of functions and quadrature formulas

It is investigated the question about an order of convergence on concrete functions, defined in s-dimensional cube $G={x=(x_1,dots,x_s): 0s,$ complicated cubature formulas. Denote $l$ a functional of errors of the cubature formula $$ (l,f)=intlimits_{G}f(x)dx-sum_{k=1}^{N}c_k f(x_k), $$ where all $c_k$ are constants, $x_k$ are points from closure $G,$ equal $0$ on polynomials of degree less than $m.$ By $n$ will denote natural numbers, by $gamma=(gamma_1,dots,gamma_n)$ --- integer vectors. Assume $h=n^{-1}; l^h$functionals of errors of complicated cubature formulas of integration on $G$ $$ l^h(x)=sum_{gamma_1=0}^{n-1}cdotssum_{gamma_s=0}^{n-1}l left(frac{x}{h}-gammaright). $$ The next statement for one-dimensional case was proved previously in [1]. newtheorem{ter}{Theorem} begin{ter} In order that for every function $fin W_p^{(m)}(G)$ fullfilled $$ (l^h,f)=o(h^m) qquad for ; hto0 $$ it is necessary and sufficiently that $l$ will equal $0$ for all polynomials of degree $m.$ end{ter} Theorem 1 and theorem 5 [2] are generalized on cubature formulas with bounded domain of integration $Gamma.$ $Gamma$ is not necessary equal $G.$ begin{thebibliography}{10} bibitem{1} S. M. Nickolsky Quadrature formulas. M.: Nauka, 1979. bibitem{2} V. I. Polovinkin Convergence of sequences of cubature formulas with boundary layer on concrete functions // Mathematical analysis and adjacent questions of mathematics. Novosibirsk, Nauka, 1978. P.~183--191. rasnoyarsk, 1994. end{thebibliography}

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