On optimization of cubature formulae for Sobolev classes of functions defined on star domains

TL;DR

This paper develops asymptotically optimal methods for numerical integration of multivariate functions on star domains, extending previous convex domain results to more general star-shaped regions with bounded gradients.

Contribution

It generalizes the optimization of cubature formulas for Sobolev classes from convex to star domains, providing new asymptotic optimality results.

Findings

Derived asymptotically optimal cubature formulas for star domains.

Extended known convex domain results to more general star-shaped regions.

Provided theoretical bounds for the recovery of the integration operator.

Abstract

We find asymptotically optimal methods of recovery of the integration operator given values of the function at a finite number of points for a class of multivariate functions defined on a bounded star domain that have bounded in $L_p$ norm of their distributional gradient; thus we generalize the known solution of this optimization problem in the case, when the domain of definition of the functions is convex.