Faber Series for \(L^2\) Holomorphic One-Forms on Riemann Surfaces with Boundary

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