Cubature formulae and polynomial ideals
The main purpose of this paper is to study the structure of cubature formulae using the notion of a polynomial ideal and its variety. The author proves that if \(I\) is a polynomial ideal generated by a proper set of \((2n-1)\) orthogonal polynomials and if the cardinality of the variety \(V(I)\) is equal to the codimension of \(I\), then there exists a cubature formula of degree \(2n-1\) based on the points in the variety. The result generalizes Gaussian cubature formulae. This result also offers a new method for constructing cubature formulae.
scientific article; zbMATH DE number 1420092
scientific article