Behavior of polynomials of best uniform approximation

Abstract

We investigate the asymptotic behavior of the polynomials { P n ( f ) } 0 ∞ \{ {P_n}(f)\} _0^\infty of best uniform approximation to a function f f that is continuous on a compact set K K of the complex plane C {\mathbf {C}} and analytic in the interior of K K , where K K has connected complement. For example, we show that for "most" functions f f , the error f − P n ( f ) f - {P_n}(f) does not decrease faster at interior points of K K than on K K itself. We also describe the possible limit functions for the normalized error ( f − P n ( f ) ) / E n (f - {P_n}(f))/{E_n} , where E n := | | f − P n ( f ) | | K {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 f on K K exist that converge more rapidly at the interior points of K K .