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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 Holmes, R., Geometric Functional Analysis and its Applications. Graduate Texts in Mathematics 24, Springer-Verlag. 1975.
 Sobolev, S.L. and Vaskevich, V.L. The Theory of Cubature Formulas. Kluwer Academic Publishers, Dordrecht, 1997.
 Kutateladze S.S., Fundamentals of Functional Analysis., Kluwer texts in the Math. Sciences: Volume 12, Kluwer Academic Publishers, Dordrecht, 1996, 229 pp.
 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.
 Bezhaev, A.Yu. and Vasilenko, V.A. Variational Spline Theory. Bull. of Novosibirsk Computing Center. Series: Numerical Analysis. Special Issue: 3. 1993.
 Holmes, R., A Course on Optimization and Best Approximation., Lecture Notes in Math. 257, Springer-Verlag. 1972.