Optimal cubature formulas in a reflexive Banach space

V. Vaskevich

Open Mathematics (2003)

  • Volume: 1, Issue: 1, page 79-85
  • ISSN: 2391-5455

Abstract

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Sequences of cubature formulas with a joint countable set of nodes are studied. Each cubature formula under consideration has only a finite number of nonzero weights. We call a sequence of such kind a multicubature formula. For a given reflexive Banach space it is shown that there is a unique optimal multicubature formula and the sequence of the norm of optimal error functionals is monotonically decreasing to 0 as the number of the formula nodes tends to infinity.

How to cite

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V. Vaskevich. "Optimal cubature formulas in a reflexive Banach space." Open Mathematics 1.1 (2003): 79-85. <http://eudml.org/doc/268801>.

@article{V2003,
abstract = {Sequences of cubature formulas with a joint countable set of nodes are studied. Each cubature formula under consideration has only a finite number of nonzero weights. We call a sequence of such kind a multicubature formula. For a given reflexive Banach space it is shown that there is a unique optimal multicubature formula and the sequence of the norm of optimal error functionals is monotonically decreasing to 0 as the number of the formula nodes tends to infinity.},
author = {V. Vaskevich},
journal = {Open Mathematics},
keywords = {65D32; 41A55; 46N05},
language = {eng},
number = {1},
pages = {79-85},
title = {Optimal cubature formulas in a reflexive Banach space},
url = {http://eudml.org/doc/268801},
volume = {1},
year = {2003},
}

TY - JOUR
AU - V. Vaskevich
TI - Optimal cubature formulas in a reflexive Banach space
JO - Open Mathematics
PY - 2003
VL - 1
IS - 1
SP - 79
EP - 85
AB - Sequences of cubature formulas with a joint countable set of nodes are studied. Each cubature formula under consideration has only a finite number of nonzero weights. We call a sequence of such kind a multicubature formula. For a given reflexive Banach space it is shown that there is a unique optimal multicubature formula and the sequence of the norm of optimal error functionals is monotonically decreasing to 0 as the number of the formula nodes tends to infinity.
LA - eng
KW - 65D32; 41A55; 46N05
UR - http://eudml.org/doc/268801
ER -

References

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  1. [1] Holmes, R., Geometric Functional Analysis and its Applications. Graduate Texts in Mathematics 24, Springer-Verlag. 1975. Zbl0336.46001
  2. [2] Sobolev, S.L. and Vaskevich, V.L. The Theory of Cubature Formulas. Kluwer Academic Publishers, Dordrecht, 1997. Zbl0877.65009
  3. [3] Kutateladze S.S., Fundamentals of Functional Analysis., Kluwer texts in the Math. Sciences: Volume 12, Kluwer Academic Publishers, Dordrecht, 1996, 229 pp. 
  4. [4] Vaskevich, V.L., Best approximation and hierarchical bases. Selçuk Journal of Applied Mathematics. 2001. V. 2, No. 1, P. 83–106. The full text version of the article is available via http://www5.in.tum.de/selcuk/sjam012207.html. 
  5. [5] Bezhaev, A.Yu. and Vasilenko, V.A. Variational Spline Theory. Bull. of Novosibirsk Computing Center. Series: Numerical Analysis. Special Issue: 3. 1993. 
  6. [6] Holmes, R., A Course on Optimization and Best Approximation., Lecture Notes in Math. 257, Springer-Verlag. 1972. Zbl0235.41016

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