Minimal cubature rules and Koornwinder polynomials

Abstract

In his classical paper (Koornwinder, 1974), Koornwinder studied a family of orthogonal polynomials of two variables, derived from symmetric polynomials. This family possesses a rare property that orthogonal polynomials of degree $n$ have $n(n+1)/2$ real common zeros, which leads to important examples in the theory of minimal cubature rules. This paper aims to give an account of the minimal cubature rules of two variables and examples originating from Koornwinder polynomials, and we will also provide further examples.