Weighted rational approximation in the complex plane

Given a triple (G, W: gamma) of an open bounded set G in the complex plane, a weight function W(z) which is analytic and different from zero in G, and a number gamma with 0 less than or equal to gamma less than or equal to 1, we consider the problem of locally uniform rational approximation of any f unction f(z), which is analytic in G, by weighted rational functions {Wmi+n i (z)R-mi,R-ni(z)}(i=0)(infinity) where R-mi,R-ni (z) = P-mi(z)/Q(ni)(z) wi th deg P-mi less than or equal to m(i) and deg Q(ni) less than or equal to n(i) for all i greater than or equal to 0 and where m(i) + n(i) --> infinit y as i --> infinity such that lim(i-->infinity) m(i)/[m(i) + n(i)] = gamma. Our main result is a necessary and sufficient condition for such an approx imation to be valid. Applications of the result to various classical weight s are also included. (C) Elsevier, Paris.