Weighted optimal order of convergence cubature formulas in Sobolev space L¯P(m) (Kn).

The main area of application of various spaces of generalized functions lies in the theory of differential equations and in the theory of quadrature and cubature formulas. Therefore, it becomes necessary to study spaces of generalized functions, one way or another related to various domains in Rⁿ. The theory of differential equations in the space of generalized functions differs from the theory of these equations in the space of ordinary functions. Deriving these equations and finding their solutions are important in applications. In the study of various questions arising in the theory of approximate integration and partial differential equations and related departments of analysis, the so-called Functional approach turned out to very fruitful. In this paper we investigate weighted cubature formula in the functional spaces L ¯ P (m) of S.L. Sobolev for the functions defined in the n - dimensional unit cube K_n and obtain an upper estimate for the norm of error functionals of weighted cubature formulas. Based on the Bakhvalov theorem it is proved that considered cubature formulas of the optimal on order of convergence in these spaces. [ABSTRACT FROM AUTHOR]