Optimal cubature formulas for calculation of multidimensional integrals in weighted Sobolev spaces

Abstract

Optimal cubature formulas are constructed for calculations of multidimensional integrals in weighted Sobolev spaces. We consider some classes of functions defined in the cube Ω = [-1, 1]^l, l = 1, 2,..., and having bounded partial derivatives up to the order r in Ω and the derivatives of jth order (r < j ≤ s) whose modulus tends to infinity as power functions of the form (d(x, Г))^-(j-r), where x ∈ Ω Г, x = (x₁,..., x_l), Г = ∂Ω, and d(x, Г) is the distance from x to Г.