Minimal Cubature rules and polynomial interpolation in two variables

TL;DR

This paper constructs minimal cubature rules of degree 4n-1 for specific weight functions on a square, relates them to Gaussian cubature rules, and develops associated polynomial interpolation with Lebesgue constant analysis.

Contribution

It explicitly constructs minimal cubature rules for complex weight functions and links them to Gaussian rules, also analyzing interpolation properties and Lebesgue constants.

Findings

Explicit minimal cubature rules of degree 4n-1 are derived.

Lagrange interpolation polynomials are constructed on the cubature nodes.

Lebesgue constants for the interpolation are determined.

Abstract

Minimal cubature rules of degree $4n-1$ for the weight functions $$ W_{\a,\b,\pm \frac12}(x,y) = |x+y|^{2\a+1} |x-y|^{2\b+1} ((1-x^2)(1-y^2))^{\pm \frac12} $$ on $[-1,1]^2$ are constructed explicitly and are shown to be closed related to the Gaussian cubature rules in a domain bounded by two lines and a parabola. Lagrange interpolation polynomials on the nodes of these cubature rules are constructed and their Lebesgue constants are determined.