Minimal Cubature rules and polynomial interpolation in two variables II
For researchers in numerical integration and approximation theory, this provides theoretical and constructive results for minimal cubature rules, though the scope is specialized.
The paper proves existence of minimal cubature rules of degree 4m+1 for specific weight functions on [-1,1]^2 and constructs near-minimal rules with one extra node, along with studying Lagrange interpolation on those nodes.
As a complement to \cite{X12}, minimal cubature rules of degree $4m+1$ for the weight functions $$ \mathcal{W}_{α,β,\pm \frac12}(x,y) = |x+y|^{2α+1} |x-y|^{2β+1} ((1-x^2)(1-y^2))^{\pm \frac12} $$ 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.