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

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