Rational approximation with varying weights in the complex plane

Given an open bounded set G in the complex plane and a weight function W (z) which is analytic and di erent from zero in G, we consider the problem of locally uniform rational approximation of any function f(z), which is analytic in G, by particular ray sequences of weighted rational functions of the form Wm+n(z)Rm;n(z), where Rm;n(z) = Pm(z)=Qn(z); with deg Pm m and degQn n: The main result of this paper is a necessary and su cient condition for such an approximation to be valid. We also consider a number of applications of this result to various classical weights, and nd explicit criteria for the possibility of weighted approximation in these cases. x