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 \(\Sigma \).
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The authors are grateful to the referees for their valuable comments.
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Communicated by Igor Pritsker
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Eric Schippers acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC).
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Schippers, E., Shirazi, M. Faber Series for \(L^2\) Holomorphic One-Forms on Riemann Surfaces with Boundary. Comput. Methods Funct. Theory (2024). https://doi.org/10.1007/s40315-024-00529-4
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DOI: https://doi.org/10.1007/s40315-024-00529-4