Let $\Omega\subset\mathbb{C}^n$ be any $C^{\infty}$ bounded domain and $H^2(\partial\Omega)$ denotes a Holomorphic Hardy space which is a $L^2(\partial\Omega)$ closure of $A^{\infty}(\Omega)(=\mathscr{O}(\Omega)\cap C^{\infty}(\overline{\Omega})).$ Because $H^2(\partial\Omega)$ is a closed subspace of  $L^2(\partial\Omega)$ there exists a countable dense subset {$\phi_j$}  where $ j\in\mathbb{N}$ such that $\forall f\in H^2(\partial\Omega)$, $f=\sum_{j=1}^{\infty}<f,\phi_j>\phi_j$. 

My Question is can we choose $ \forall j\in\mathbb{N},\;\; \phi_j\in A^{\infty}(\Omega)?$