On the condition number of some gram matrices arising from least squares approximation in the complex plane

This paper analyses the growth of the condition number of a class of Gram matrices that arise when computing least squares polynomials in polygons of the complex plane. It is shown that if the polygon is inserted between two ellipses then the condition number of the (n+1)×(n+1) Gram matrix is bounded from above by 4 m(n+1)²(k)^{2 n} where m is the number of edges of the polygon, and k≥1 is a known ratio which is close to one if the two ellipses are close to each other.