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 f unction f(z), which is analytic in G, by weighted polynomials of the f orm (W-n(z)) P-n (z)(n=0)(infinity), where deg P-n less than or equal to n. The main result of this paper is a necessary and sufficient cond ition for such an approximation to be valid. We also consider a number of applications of this result to various classical weights, which gi ve explicit criteria for these weighted approximations.