Abrahamse's interpolation theorem and Fuchsian groups

TL;DR

This paper extends Abrahamse's interpolation theorem to more general Riemann surfaces associated with Fuchsian groups, using duality techniques and kernel function relations, with applications to specific Fuchsian group actions.

Contribution

It generalizes Abrahamse's theorem to Fuchsian groups and develops new duality-based interpolation methods on Riemann surfaces.

Findings

Established a scalar-valued interpolation theorem for Fuchsian group automorphic functions.

Proved a generalized distance formula similar to Nehari's theorem.

Analyzed kernel functions in relation to Szegő kernels for disk Fuchsian groups.

Abstract

We generalize Abrahamse's interpolation theorem from the setting of a multiply connected domain to that of a more general Riemann surface. Our main result provides the scalar-valued interpolation theorem for the fixed-point subalgebra of $H^\infty$ associated to the action of a Fuchsian group. We rely on two results from a paper of Forelli. This allows us to prove the interpolation result using duality techniques that parallel Sarason's approach to the interpolation problem for $H^\infty$. In this process we prove a more general distance formula, very much like Nehari's theorem, and obtain relations between the kernel function for the character automorphic Hardy spaces and the Szeg\"o kernel for the disk. Finally, we examine our interpolation results in the context of the two simplest examples of Fuchsian groups acting on the disk.