Can any conformal mapping on a bounded multiply-connected domain be approximated by polynomials?

Question

Mergelyan's theorem states that a necessary condition for a function, which is analytic inside a compact set $X$ in the complex plane $\Bbb C$ and continuous on the boundary of $X$, to be uniformly approximated by polynomials is that the complement of $X$ in $\Bbb C$ is connected. I want to know that whether conformal mappings on $X$ have a similar property and whether they can always be uniformly approximated by polynomials. Thank you!

Answer 1

Let $\mathbb{CP}^1$ be the Riemannian sphere, $K \subseteq \mathbb{C}$ a compact set, and $\mathcal{H}(K)$ the family of holomorphic functions on $K$.

Let $E\subseteq \mathbb{CP}^1 \setminus K$ be a set (which will be the set of poles), and let $\mathcal{R}(E)$ be the set of rational functions with poles at $E$.

Runge's Theorem states that $\mathcal{R}(E)$ is dense in $\mathcal{H}(K)$ if, and only if, $E$ cuts every connected component of $\mathbb{CP}^1 \setminus K$.

*Here convergence is with some metric.

**The fact that $\mathcal{R}(E)$ is dense in $\mathcal{H}(K)$ implies that every holomorphic function $f \in \mathcal{H}(K)$ can be approximated by rational functions with poles at $E$.

If the complement of $K$ in $\mathbb{C}$ (or $\mathbb{CP}^1$) is connected, one can take $E =\{\infty\}$, which cuts the unique connected component of $\mathbb{CP}^1\setminus K$. By Runge's Theorem we obtain that every holomorphic function on $K$ can be approximated by rational functions with poles at $\{\infty\}$, i.e. polynomials.

Conformal mappings are holomorphic, so this also implies that you can approximate conformal mappings on $K$ by polynomials (and this approximation is the usual uniform convergence).