Sequences with equi-distributed extreme points in uniform polynomial approximation

Abstract

Let E be a compact set in C with connected complement and positive logarithmic capacity. For any f continuous on E and analytic in the interior of E, we consider the distribution of extreme points of the error of best uniform polynomial approximation on E. Let Λ=( n j ) be a subsequence of N such that n j+1 / n j →1. If, for n∈ Λ, A n( f)⊆ ∂E denotes the set of extreme points of the error function, we prove that there is a subsequence Λ′ of Λ such that the distribution of any ( n+2)th Fekete point set F n+2 of A n( f) tends weakly to the equilibrium distribution on E as n→∞ in Λ′. Furthermore, we prove a discrepancy result for the distribution of the point sets F n+2 if the boundary of E is smooth enough.