Bivariate interpolation at Xu points: results, extensions and applications

In a recent paper, Y. Xu proposed a set of Chebyshev-like points for polynomial interpolation on the square $[-1,1]^2$. We have recently proved that the Lebesgue constant of these points grows like $\log^2$ of the degree (as with the best known points for the square), and we have implemented an accurate version of their Lagrange interpolation formula at linear cost. Here we construct non-polynomial Xu-like interpolation formulas on bivariate compact domains with various geometries, by means of composition with suitable smooth transformations. Moreover, we show applications of Xu-like interpolation to the compression of surfaces given as large scattered data sets.