Информационная система "Конференции"
Международная конференция по вычислительной математике МКВМ-2002
- 24-28 июня 2002 г.
- Новосибирск, Академгородок
Тезисы докладов
Аппроксимация функций и квадратурные формулы
Сходимость усложненных кубатурных формул на конкретных функциях
Красноярский гос. техн. ун-т (Красноярск)
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}
Примечание. Тезисы докладов публикуются в авторской редакции
© 1996-2000, Институт вычислительных технологий СО РАН, Новосибирск
© 1996-2000, Сибирское отделение Российской академии наук, Новосибирск
Дата последней модификации: 06-Jul-2012 (11:45:20)