Minimal Cubature rules and polynomial interpolation in two variables II

TL;DR

This paper extends the construction of minimal and near minimal cubature rules of degree 4m+1 for specific weight functions on the square, and studies associated Lagrange interpolation polynomials.

Contribution

It introduces new minimal and near minimal cubature rules for degree 4m+1, expanding previous work, and analyzes the interpolation polynomials on these nodes.

Findings

Existence of minimal cubature rules for specified weight functions.

Explicit construction of near minimal cubature rules with one extra node.

Analysis of Lagrange interpolation polynomials on the cubature nodes.

Abstract

As a complement to \cite{X12}, minimal cubature rules of degree $4m+1$ for the weight functions $$ \mathcal{W}_{\alpha,\beta ,\pm \frac12}(x,y) = |x+y|^{2\alpha+1} |x-y|^{2\beta+1} ((1-x^2)(1-y^2))^{\pm \frac12} $$ on $[-1,1]^2$ are shown to exist and near minimal cubature rules of the same degree with one node more than minimal are constructed explicitly. The Lagrange interpolation polynomials on the nodes of the near minimal cubature rules are also studied.