The distribution of extreme points in best complex polynomial approximation

Abstract

LetK be a compact point set in the complex plane having positive logarithmic capacity and connected complement. For anyf continuous onK and analytic in the interior ofK we investigate the distribution of the extreme points for the error in best uniform approximation tof onK by polynomials. More precisely, if $$A_n (f): = \{ z \in K:|f(z) - p_n^* (f;z)| = \parallel f - p_n^* (f)\parallel _K \} ,$$ wherep n * (f) is the polynomial of degree≤n of best uniform approximation tof onK, we show that there is a subsequencenk with the property that the sequence of (n k +2)-point Fekete subsets of\(A_{n_k }\) has limiting distribution (ask→∞) equal to the equilibrium distribution forK. Analogues for weighted approximation are also given.