Optimal cubature formulas in a reflexive Banach space

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


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.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [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.


Journal + Issues

Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant and original works in all areas of mathematics. The journal publishes both research papers and comprehensive and timely survey articles. Open Math aims at presenting high-impact and relevant research on topics across the full span of mathematics.