Smooth equilibrium measures and approximation

A necessary and sufficient condition is given for approximation with weighted expressions of the form wnPn, where w is a given continuous weight function and Pn are polynomials of degree . The condition is that the extremal measure that solves an associated equilibrium problem is smooth (asymptotically optimal doubling). As corollaries we get all previous (positive and negative) results for approximation, as well as the solution of a problem of T. Bloom and M. Branker. A connection to level curves of homogeneous polynomials of two variables is also explored.