Show that any holomorphic function \(f\) on a simply connected domain (or compact set with connected complement) can be uniformly approximated by polynomials on compact subsets.
Solution
There are two standard forms of the statement.
1. Let \(K\subset\mathbb C\) be compact with connected complement, and suppose \(f\) is holomorphic on a neighborhood of \(K\). By Runge’s theorem, for every \(\varepsilon>0\) there exists a polynomial \(p\) such that\[ \sup_{z\in K}|f(z)-p(z)|<\varepsilon. \]Indeed, Runge gives approximation by rational functions whose poles may be chosen in the components of \(\widehat{\mathbb C}\setminus K\). Since this complement is connected, the only required pole may be placed at \(\infty\); rational functions with their only pole at \(\infty\) are polynomials.
2. Now let \(\Omega\subset\mathbb C\) be simply connected and let \(f\in H(\Omega)\). Fix a compact set \(K\Subset\Omega\). Choose a bounded Jordan domain \(U\) such that\[ K\subset U,\qquad \overline U\subset\Omega. \]Such a \(U\) exists because a simply connected plane domain admits an exhaustion by relatively compact Jordan domains. The compact set \(L=\overline U\) has connected complement, and \(f\) is holomorphic on a neighborhood of \(L\).
Applying the compact form of Runge’s theorem to \(L\), for every \(\varepsilon>0\) there is a polynomial \(p\) with \[ \sup_{z\in L}|f(z)-p(z)|<\varepsilon. \] Since \(K\subset L\), \[ \sup_{z\in K}|f(z)-p(z)|<\varepsilon. \] Thus polynomials approximate \(f\) uniformly on every compact subset of \(\Omega\).