  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.
