Optimal Points for Cubature Rules and Polynomial Interpolation on a Square

TL;DR

This paper explores optimal node points for cubature rules and polynomial interpolation on a square, focusing on their distribution, orthogonal polynomials, and convergence properties, with insights into the underlying theory.

Contribution

It analyzes the theory of minimal cubature nodes and their relation to orthogonal polynomials specifically on square domains, highlighting their distribution and interpolation properties.

Findings

Nodes are common zeros of orthogonal polynomials of degree n.

Nodes are well distributed and lead to desirable convergence in interpolation.

The paper summarizes known results and theoretical insights for square domains.

Abstract

The nodes of certain minimal cubature rule are real common zeros of a set of orthogonal polynomials of degree $n$. They often consist of a well distributed set of points and interpolation polynomials based on them have desired convergence behavior. We report what is known and the theory behind by explaining the situation when the domain of integrals is a square.