Remarks on “almost best” Approximation in the Complex Plane

Abstract

Let f(x) be a continuous function on the compact interval J of the real axis, which is not the restriction of a function holomorphic in a neighborhood of J. Let π _n be the set of all polynomials over ℂ of degree ≤ n. Let p _n (x) be the polynomial of best approximation to f(x) on J, i.e.,

$$ E_n(f,J)=in f_{q \in {\pi_{n}} } \parallel f-q \parallel J= \parallel f-p_n\parallel J, $$

where the norm is the sup-norm.