Uniform Approximation by Polynomials

Extend Mergelyan's theorem to the case in which $S^{2}-K$ has finitely many components: Prove that then every $f \in C(K)$ which is holomorphic in the interior of $K$ can be uniformly approximated on $K$ by rational functions.

Show that the result of Exercise 1 does not extend to arbitrary compact sets $K$ in the plane, by verifying the details of the following example. For $n = 1,2,3, \ldots$, let $D_{n} = D\left(\alpha_{n} ; r_{n}\right)$ be disjoint open discs in $U$ whose union $V$ is dense in $U$, such that $\Sigma r_{n} < \infty$. Put $K = \bar{U}-V$. Let $\Gamma$ and $\gamma_{n}$ be the paths

and define

Prove that $L$ is a bounded linear functional on $C(K)$, prove that $L(R) = 0$ for every rational function $R$ whose poles are outside $K$, and prove that there exists an $f \in C(K)$ for which $L(f) \neq 0$.

Show that the function $g$ constructed in the proof of Lemma 20.2 has the smallest supremum norm among all $f \in H(\Omega )$ such that $z f(z) \rightarrow 1$ as $z \rightarrow \infty$. (This motivates the proof of the lemma).

Show also that $b = c_{0}$ in that proof and that the inequality $\left|b\right| < 4 r$ can therefore be replaced by $\left|b\right| < r$. In fact, $b$ lies in the convex hull of the set $E$.