Optimal cubature formulas in a reflexive Banach space

V. Vaskevich 1
  • 1 Siberian Division of Russian Academy of Sciences

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.

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  • [1] Holmes, R., Geometric Functional Analysis and its Applications. Graduate Texts in Mathematics 24, Springer-Verlag. 1975.

  • [2] Sobolev, S.L. and Vaskevich, V.L. The Theory of Cubature Formulas. Kluwer Academic Publishers, Dordrecht, 1997.

  • [3] Kutateladze S.S., Fundamentals of Functional Analysis., Kluwer texts in the Math. Sciences: Volume 12, Kluwer Academic Publishers, Dordrecht, 1996, 229 pp.

  • [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] Bezhaev, A.Yu. and Vasilenko, V.A. Variational Spline Theory. Bull. of Novosibirsk Computing Center. Series: Numerical Analysis. Special Issue: 3. 1993.

  • [6] Holmes, R., A Course on Optimization and Best Approximation., Lecture Notes in Math. 257, Springer-Verlag. 1972.

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