Weighted polynomial approximation in the complex plane

Given a pair $(G,W)$ of an open bounded set $G$ in the complex plane and a weight function $W(z)$ which is analytic and different from zero in $G$, we consider the problem of the locally uniform approximation of any function $f(z)$, which is analytic in $G$, by weighted polynomials of the form $\left \{W^{n}(z)P_{n}(z) \right \}^{\infty }_{n=0}$, where $\deg P_{n} \leq n$. The main result of this paper is a necessary and sufficient condition for such an approximation to be valid. We also consider a number of applications of this result to various classical weights, which give explicit criteria for these weighted approximations.