Minimal Cubature rules and polynomial interpolation in two variables
For researchers in numerical integration and approximation theory, this provides new explicit cubature rules and interpolation results for a class of weight functions.
The paper constructs explicit minimal cubature rules of degree 4n-1 for specific weight functions on [-1,1]^2 and shows their connection to Gaussian cubature rules. It also constructs Lagrange interpolation polynomials on these nodes and determines their Lebesgue constants.
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.