Uniqueness of the Gaussian Quadrature for a Ball Original Research Article

We construct a formula for numerical integration of functions over the unit ball in Rd that uses n Radon projections of these functions and is exact for all algebraic polynomials in Rd of degree 2n−1. This is the highest algebraic degree of precision that could be achieved by an n term integration rule of this kind. We prove the uniqueness of this quadrature. In particular, we present a quadrature formula for a disk that is based on line integrals over n chords and integrates exactly all bivariate polynomials of degree 2n−1.