Abstract
We investigate the asymptotic behavior of the polynomials $\{ {P_n}(f)\} _0^\infty$ of best uniform approximation to a function $f$ that is continuous on a compact set $K$ of the complex plane ${\mathbf {C}}$ and analytic in the interior of $K$, where $K$ has connected complement. For example, we show that for "most" functions $f$, the error $f - {P_n}(f)$ does not decrease faster at interior points of $K$ than on $K$ itself. We also describe the possible limit functions for the normalized error $(f - {P_n}(f))/{E_n}$, where ${E_n}: = ||f - {P_n}(f)|{|_K}$, and the possible limit distributions of the extreme points for the error. In contrast to these results, we show that "near best" polynomial approximants to $f$ on $K$ exist that converge more rapidly at the interior points of $K$.
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