Minimal cubature rules and Koornwinder polynomials

TL;DR

This paper explores minimal cubature rules in two variables derived from Koornwinder polynomials, highlighting their unique zero properties and providing new examples in the field.

Contribution

It introduces new minimal cubature rules based on Koornwinder polynomials and expands the set of known examples in multivariate numerical integration.

Findings

Koornwinder polynomials have n(n+1)/2 common zeros for degree n

New minimal cubature rules are constructed from these polynomials

Additional examples of cubature rules are provided

Abstract

In his classical paper [5], 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.