Nontensorial Clenshaw-Curtis cubature

We extend Clenshaw-Curtis quadrature to the square in a nontensorial way, by using Sloan’s hyperinterpolation theory and two families of points recently studied in the framework of bivariate (hyper)interpolation, namely the Morrow-Patterson-Xu points and the Padua points. The con- struction is an application of a general approach to product-type cubature, where we prove also a relevant stability theorem. The resulting cubature formulas turn out to be competitive on nonentire integrands with tensor- product Clenshaw-Curtis and Gauss-Legendre formulas, and even with the few known minimal formulas.