Weighted rational approximation in the complex plane

Abstract

Given a triple ( G, W, γ) 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 γ with 0 ≤ γ ≤ 1, we consider the problem of locally uniform rational approximation of any function ƒ( z), which is analytic in G, by weighted rational functions W m i + n i ( z) R m i , n i ( z) i = 0 ∞, where R m i, n i( z) = P m i ( z)/ Q n i ( z) with deg P m i ≤ m i and deg Q n i ≤ n i for all i ≥ 0 and where m i + n i → ∞ as i → ∞ such that lim m i /[ m i + n i ] = γ. Our main result is a necessary and sufficient condition for such an approximation to be valid. Applications of the result to various classical weights are also included.