Interpolation points and interpolation formulae on the square

The problem of choosing " good " nodes on a given compact set is a central one in multivariate polynomial interpolation. Besides unisolvence, which is by no means an easy problem, for practical purposes one needs slow growth of the Lebesgue constant and computational efficiency. In this talk, we present new sets of nodes recently studied for polynomial interpolation on the square that are asymptotically equally-spaced w.r.t. the Dubiner metric. Then, we shall deal in particular with two families, Xu and Padua points, which show Lebesgue constants with optimal grow, i.e. log 2 (n), with n the degree of the interpolating polynomial. Finally, we discuss some applications of such points.