The paper aims to estimate the error of a cubature formula acting on an arbitrary function from the Sobolev space on a multidimensional cube. The norm of the error in the dual space of the Sobolev class is represented as a positive definite quadratic form in the weights of the cubature formula. Given a finite smoothness s of the Sobolev space, we establish a power-law convergence order to zero of error functionals norms for near minimal cubature formulas and cubature formulas of high trigonometric degree. If the smoothness s of the Sobolev space tends to infinity then the power-law convergence order tends to infinity as well.
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