A Carleman type theorem for proper holomorphic embeddings
Gregery T. Buzzard and Franc Forstneric
$ \def\C{{\mathbb C}} \def\cC{{\mathcal C}} $
In 1927, Carleman showed that a continuous, complex-valued function on the real line can be approximated in the Whitney topology by an entire function restricted to the real line. In this paper, we prove a similar result for proper holomorphic embeddings. Namely, we show that a proper $\cC^r$ embedding of the real line into $\C^n$ can be approximated in the strong $\cC^r$ topology by a proper holomorphic embedding of $\C$ into $\C^n$.