The infinite convergence order of near minimal cubature formulas on classes of periodic functions

ABSTRACT

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.

KEYWORDS:

• Cubature formulas
• near minimal cubature formulas
• infinite convergence order
• error functional
• extremal function
• embedding constant

AMS SUBJECT CLASSIFICATIONS:

• 65D32
• 41A44
• 41A63
• 31B30

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Correction

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no. 0314-2019-0013).