Faber Series for $$L^2$$ Holomorphic One-Forms on Riemann Surfaces with Boundary

Abstract

Consider a compact surface \(\mathscr {R}\) with distinguished points \(z_1,\ldots ,z_n\) and conformal maps \(f_k\) from the unit disk into non-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 Faber polynomials 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 series converges both in \(L^2(\Sigma )\) and uniformly on compact subsets of \(\Sigma \).