A Rudin-Carleson theorem with uniform approximation for manifold-valued maps

Given a closed set $E \subset \partial {\mathbb D}$ of measure zero and a continuous function $\varphi : E \to {\mathbb C}$, the classical Rudin-Carleson interpolation theorem states that there exists a continuous function $F : \overline {\mathbb D} \to {\mathbb C}$ that is holomorphic on ${\mathbb D}$ and satisfies $F\rvert_E = \varphi$. In this paper we obtain a generalisation of the Rudin-Carleson theorem for maps $\varphi : E \to X$ into arbitrary connected complex manifolds $X$ that combines interpolation of $\varphi$ on $E$ with uniform approximation on compact subsets of $\overline {\mathbb D} \setminus E$ of another given continuous map $f:\overline {\mathbb D} \to X$ that is holomorphic on ${\mathbb D}$. Under the further assumption that $X$ is an Oka manifold we obtain a corollary that combines Rudin-Carleson interpolation of a continuous map $\varphi : \overline {\mathbb D} \to X$ on $E$ with Runge approximation of $\varphi$ on a compact set $K\subset {\mathbb D}$ without any holes on which $\varphi$ is holomorphic.