Bivariate polynomial interpolation on the square at new nodal sets

Abstract

As known, the problem of choosing “good” nodes is a central one in polynomial interpolation. While the problem is essentially solved in one dimension (all good nodal sequences are asymptotically equidistributed with respect to the arc-cosine metric), in several variables it still represents a substantially open question. In this work we consider new nodal sets for bivariate polynomial interpolation on the square. First, we consider fast Leja points for tensor-product interpolation. On the other hand, for interpolation in P n 2 on the square we experiment four families of points which are (asymptotically) equidistributed with respect to the Dubiner metric, which extends to higher dimension the arc-cosine metric. One of them, nicknamed Padua points, gives numerically a Lebesgue constant growing like log square of the degree.