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International Conference on Mathematical Modeling and Scientific Computing
April 2-6, 2001
Middle East Technical University and Selcuk University
Ankara and Konya, Turkey

Organizers
F. Bornemann (Munich University of Tecnology, Germany), H. Bulgak (Selcuk University, Konya, Turkey), V. Ganzha (Munich University of Technology, Germany), B. Karasozen (METU, Ankara, Turkey), A. Sinan (Selcuk University, Konya, Turkey), C. Zenger (Munich University of Technology, Germany)

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Computation of Integrals with Guaranteed Accuracy and Bases in Sobolev spaces
by
Vladimir Vaskevitch
Sobolev Institute of Mathematics, SB RAS, Novosibirsk, Russia

Let \omega(x) be a continuous weight function with domain \Omega; \Omega subset Rn. We consider the sequence of the weighted cubature formulas
(lN, j)=
ó
õ
\Omega 
 \omega(x)j(x) dx- N
å
k=1 
 ck j(xk) \approx 0,     N=1, 2, ....
To the error lN we assign the set KN(\epsilon) by putting
KN(\epsilon)={j in C(
\Omega
 
 ) | |(lN, j)| <= \epsilon},     \epsilon > 0,
with C([`(\Omega)]) the well-known Banach space of continuous functions. By definition, the Nth cubature formula under consideration enables us to calculate the integral \int\Omega\omega(x)j(x) dx with guaranteed accuracy \epsilon iff \psi(x) is a member of KN(\epsilon). We establish the conditions which are sufficient for \psi(x) in KN(\epsilon) and discuss how to check these conditions by computing. We also consider the questions which are concerned with the construction of the bases in Sobolev spaces and closely connected with the problem of computation with guaranteed accuracy.

Date received: December 28, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagk-14.