Lagrange polynomials, reproducing kernels and cubature in two dimensions

Abstract

We obtain by elementary methods necessary and sufficient conditions for a k-dimensional cubature formula to hold for all polynomials of degree up to 2m−1 when the nodes of the formula have Lagrange polynomials of degree at most m. The main condition is that the Lagrange polynomial at each node is a scalar multiple of the reproducing kernel of degree m−1 evaluated at the node plus an orthogonal polynomial of degree m. Stronger conditions are given for the case where the cubature formula holds for all polynomials of degree up to 2m.This result is applied in one dimension to obtain a quadrature formula where the nodes are the roots of a quasi-orthogonal polynomial of order 2. In two dimensions the result is applied to obtain constructive proofs of cubature formulas of degree 2m−1 for the Geronimus and the Morrow–Patterson classes of nodes. A cubature formula of degree 2m is obtained for a subclass of Morrow–Patterson nodes. Our discussion gives new proofs of previous theorems for the Chebyshev points and the Padua points, which are special cases.