Polynomial density in Bers spaces

Abstract:

Let D be a bounded Jordan domain such that $\smallint \;\smallint {\;_D}\lambda _D^{2 - q}\;dx\;dy\; < \infty$ for $q > 1$. Here ${\lambda _D}(z)$ is the Poincaré metric for D. Define $A_q^p(D)$, the Bers space, to be the Fréchet space of holomorphic functions f on D, such that $\left \| f \right \|_{q,p}^p = \smallint \;\smallint {\;_D}\lambda _D^{2 - qp}|f{|^p}\;dx\;dy$ is finite, $0 < p < \infty ,qp > 1$. It is well known that the polynomials are dense in $A_q^p(D)$ for $qp \geqslant 2$. We show that they are dense in $A_q^p(D)$ for $qp > 1$ irrespective whether the boundary of D is rectifiable or not.