Some remarks on cubature formulas with linear operators

Abstract

In this paper we consider a novel type of cubature formulas called operator-type cubature formulas. The notion originally goes back to a famous work by G. D. Birkhoff in 1906 on Hermite interpolation problem. A well-known theorem by Sobolev in 1962 on invariant cubature formulas is generalized to operator-type cubature, which provides a systematic treatment of Lebedev's works in the 1970s and some related results by Shamsiev in 2006. We give a lower bound for the number of points needed, and discuss analytic conditions for equality, together with tight illustrations for Laplacian-type cubature.