Cubature Formulae and Polynomial Ideals

Abstract

The structure of cubature formulae of degree 2n-1 is studied from a polynomial ideal point of view. The main result states 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 covers a number of cubature formulae in the literature, including Gaussian cubature formulae on one end and the usual product formulae on the classical domains on the other end. The result also offers a new method for constructing cubature formulae. © 1999 Academic Press.