Weighted approximation for weak convex external fields

Let w ( x ) ≢ 0 be a non-negative continuous weight function which decays faster than 1 / x near infinity and let q ( x ) = − l o g w ( x ) . Totik proved that if q ( x ) is convex, then a continuous function f ( x ) can be approximated by weighted polynomials w ( x ) n P n ( x ) , n = 0 , 1 , 2 , …  , if and only if f ( x ) vanishes outside the support of the equilibrium measure associated with q ( x ) . We prove a similar result in the case when q ( x ) is only “weak convex”.