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A Rudin-Carleson theorem for multiply connected domains with interpolation

Benedikt Steinar Magnússon, Bergur Snorrason

TL;DR

The paper generalizes the Rudin–Carleson boundary-extension problem to $k$-connected domains with arclength-null boundary sets by proving an annular F. and M. Riesz theorem and leveraging a Bishop-type extension framework. The method first reduces the multiply connected case to annular and doubly connected pieces via Riemann mapping and then constructs holomorphic extensions that extend boundary data while respecting a prescribed bound on the boundary. The main contributions are Theorem 1.1 (a boundary-continuous, interior-holomorphic extension with $|F|<M$ on $\partial\Omega$) and Theorem 1.2 (an annular analogue of F. and M. Riesz), together with an interpolation-augmented version that allows prescribing interior values at finitely many points. These results extend classical boundary-extension theory to more complex domains and provide tools for interpolation in holomorphic extensions, with implications for uniform algebras on multiply connected domains.

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.

A Rudin-Carleson theorem for multiply connected domains with interpolation

TL;DR

The paper generalizes the Rudin–Carleson boundary-extension problem to -connected domains with arclength-null boundary sets by proving an annular F. and M. Riesz theorem and leveraging a Bishop-type extension framework. The method first reduces the multiply connected case to annular and doubly connected pieces via Riemann mapping and then constructs holomorphic extensions that extend boundary data while respecting a prescribed bound on the boundary. The main contributions are Theorem 1.1 (a boundary-continuous, interior-holomorphic extension with on ) and Theorem 1.2 (an annular analogue of F. and M. Riesz), together with an interpolation-augmented version that allows prescribing interior values at finitely many points. These results extend classical boundary-extension theory to more complex domains and provide tools for interpolation in holomorphic extensions, with implications for uniform algebras on multiply connected domains.

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.
Paper Structure (4 sections, 9 theorems, 18 equations, 1 figure)

This paper contains 4 sections, 9 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A Rudin-Carleson theorem for multiply connected domains
  3. Annular F. and M. Riesz theorem
  4. A Rudin-Carleson theorem with interpolation

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{C}$ be a $k$-connected domain, with $k > 1$, such that the boundary of $\Omega$ consists of $k$ pairwise disjoint Jordan curves, $E \subset \partial \Omega$ be a closed arclength null set, $f \colon E \rightarrow \mathbb{C}$ a continuous function, and $M \colon \partial \

Figures (1)

  • Figure 1: Illustrated is an example of $\Omega$ from the proof of Theorem \ref{['thm:1.1']}, were $k = 3$. The figure in the upper left shows $\Omega$, while the other three show $D_{1, 2}$, $D_{1, 3}$, and $D_{2, 3}$, depending on how the components of the boundary are numbered.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Rudin-Carleson theorem
  • Theorem 2.2: Bishop's theorem
  • Theorem 2.3: F. and M. Riesz theorem
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • proof
  • Theorem 2.6: Doubly connected Riemann mapping theorem
  • ...and 5 more