The density of extreme points in complex polynomial approximation

Abstract:

Let $K$ be a compact set in the complex plane having connected and regular complement, and let $f$ be any function continuous on $K$ and analytic in the interior of $K$. For the polynomials $p_n^*(f)$ of respective degrees at most $n$ of best uniform approximation to $f$ on $K$, we investigate the density of the sets of extreme points \[ {A_n}(f): = \{ z \in K:|f(z) - p_n^*(f)(z)| = ||f - p_n^*(f)|{|_K}\} \] in the boundary of $K$.