On approximation in the Bers spaces

Abstract:

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