Minimal cubature rules and polynomial interpolation in two variables

Minimal cubature rules of degree 4n−14n−1 for the weight functions Wα,β,±12(x,y)=|x+y|2α+1|x−y|2β+1((1−x2)(1−y2))±12 on [−1,1]2[−1,1]2 are constructed explicitly and are shown to be closely 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.