On approximation in the Bers spaces

Abstract

Let D D be a Jordan domain in the complex plane with rectifiable boundary C C . Let A q ( D ) {A_q}(D) denote the Bers space with norm | | | | q ||\;|{|_q} . We prove that if f ∈ A q ( D ) , 2 > q > ∞ f \in {A_q}(D),2 > q > \infty , then there exist functions s n ( z ) = Σ k = 1 n 1 / ( z − z n , k ) , z n , k ∈ C for k = 1 , ⋯ , n {s_n}(z) = \Sigma _{k = 1}^n1/(z - {z_{n,k}}),\;{z_{n,k}} \in C{\text { for }}k = 1, \cdots ,n , such that | | s n − f | | q → 0 ||{s_n} - f|{|_q} \to 0 . This result does not hold for 1 > q ≦ 2 1 > q \leqq 2 even when D D is a disc.