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

Consider a compact surface ℛ with distinguished points z_1,… ,z_n and conformal maps f_k from the unit disk into non-overlapping quasidisks on ℛ taking 0 to z_k . Let Σ be the Riemann surface obtained by removing the closures of the images of f_k from ℛ . We define forms which are meromorphic on ℛ with poles only at z_1,… ,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 Σ is uniquely expressible as a series of Faber–Tietz forms. This series converges both in L^2(Σ ) and uniformly on compact subsets of Σ .