A Rudin-Carleson theorem for multiply connected domains with interpolation

TL;DR

This paper generalizes the Rudin-Carleson theorem to finitely connected domains using an annular F. and M. Riesz theorem, enabling boundary-continuous holomorphic functions with finite interpolation.

Contribution

It extends the Rudin-Carleson theorem to multiply connected domains by employing an annular F. and M. Riesz theorem, allowing boundary interpolation.

Findings

Established a generalized Rudin-Carleson theorem for finitely connected domains.

Proved the existence of boundary-continuous holomorphic functions with interpolation.

Utilized an annular version of the F. and M. Riesz theorem in the proof.

Abstract

Using an annular version of the F. and M. Riesz theorem, we prove a generalization of the Rudin-Carleson theorem for finitely connected bounded domains. That is, for a continuous function on a closed set in the boundary of measure zero there is a holomorphic function on the domain continuous to the boundary. Furthermore, this can be done with interpolation at finitely many points in the domain. The proof relies on an annular version of the F. and M. Riesz theorem.