On nonlinear Rudin-Carleson type theorem

Abstract

In this paper we study nonlinear interpolation problems for interpolation and peak-interpolation sets of function algebras. The subject goes back to the classical Rudin-Carleson interpolation theorem. In particular, we prove the following nonlinear version of this theorem: Let $\bar{\mathbb{D}}\subset \mathbb{C}$ be the closed unit disk, $\mathbb{T}\subset \bar{\mathbb{D}}$ the unit circle, $S\subset \mathbb{T}$ a closed subset of Lebesgue measure zero and $M$ a connected complex manifold. Then for every continuous $M$-valued map $f$ on $S$ there exists a continuous $M$-valued map $g$ on $\bar{\mathbb{D}}$ holomorphic on its interior such that $g{|}_S=f$. We also consider similar interpolation problems for continuous maps $f:S\to \bar{M}$, where $\bar{M}$ is a complex manifold with boundary $\partial M$ and interior $M$. Assuming that $f(S)\cap \partial M\ne \varnothing$ we are looking for holomorphic extensions $g$ of $f$ such that $g(\bar{\mathbb{D}}\setminus S)\subset M$.

13 pages

References in corpus (2)

• Oka properties of complements of holomorphically convex sets
• Dense Stable Rank and Runge Type Approximation Theorems for ${\mathbf{H}}^{\infty }$ Maps