Uniform approximation of sgn(x) by polynomials and entire functions

We study the best uniform approximation by polynomials of fixed degree of the function sgn(x) on the union of two intervals symmetric with respect to the origin. We obtain precise asymptotics, with explicit constants, for the error of the best approximation as the degrees tend to infinity. Our approach is based on a new representation of the extremal polynomials in terms of certain conformal maps. We also study the best polynomial approximation of sgn(x) on [-1,1] with respect to the Hausdorff distance between the completed graphs.