Faber series for $L^2$ holomorphic one-forms on Riemann surfaces with boundary

Consider a compact surface $\mathcal{R}$ with distinguished points $z_1,\dots ,z_n$ and conformal maps $f_k$ from the unit disk into non-overlapping quasidisks on $\mathcal{R}$ taking $0$ to $z_k$. Let $\mathrm{\Sigma }$ be the Riemann surface obtained by removing the closures of the images of $f_k$ from $\mathcal{R}$. We define forms which are meromorphic on $\mathcal{R}$ with poles only at $z_1,\dots ,z_n$, which we call Faber-Tietz forms. These are analogous to Faber polynomials in the sphere. We show that any $L^2$ holomorphic one-form on $\mathrm{\Sigma }$ is uniquely expressible as a series of Faber-Tietz forms. This series converges both in $L^2(\mathrm{\Sigma })$ and uniformly on compact subsets of $\mathrm{\Sigma }$.