Faber Series for L2 Holomorphic One-Forms on Riemann Surfaces with Boundary

• Published in: Computational methods and function theory Vol. 25; no. 2; pp. 329 - 347
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.06.2025
• Subjects: Conformal mapping; Polynomials; Riemann surfaces
• ISSN: 1617-9447; 2195-3724
• DOI: 10.1007/s40315-024-00529-4

Consider a compact surface R with distinguished points z1,…,zn and conformal maps fk from the unit disk into non-overlapping quasidisks on R taking 0 to zk. Let Σ be the Riemann surface obtained by removing the closures of the images of fk from R. We define forms which are meromorphic on R with poles only at z1,…,zn, which we call Faber–Tietz forms. These are analogous to Faber polynomials in the sphere. We show that any L2 holomorphic one-form on Σ is uniquely expressible as a series of Faber–Tietz forms. This series converges both in L2(Σ) and uniformly on compact subsets of Σ.