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 . Totik proved that if q(x) is convex, then a continuous function f(x) can be approximated by weighted polynomials wn(x)Pn(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”.