The role of the endpoint in weighted polynomial approximation with varying weights

For a weight functionw: a, b]→(0, ∞), we consider weighted polynomials of the formw ⁿ P_n where the degree ofP _n is at mostn. The class of functions that can be approximated with such polynomials depends on the behavior of the densityv(t) of the extremal measure associated withw. We show that every approximable function must vanish at the endpointa ifv(t) behaves like (t?a) ^β ast→a with β>?1/2. We also present an analogous result for internal points. Our results solve some open problems posed by V. Totik and disprove a conjecture of G.G. Lorentz on incomplete polynomials.