Uniform convergence of a sequence of holomorphic polynomials

Question

This is a two-part question :

1. Let $E = \{|z|=1 \text{ and } \Im{z} \geq 0\}$. Does there exist a sequence of (holomorphic) polynomials in $z$ which converge uniformly on $E$ to $f(z) = \bar{z}$? Justify.
2. Answer the same question for the set $E=\{|z|=1\}$.

My thoughts: From Mergelyan's Theorem we know that any function holomorphic in the interior of a compact set and continuous to the boundary, can be approximated uniformly by polynomials in the compact set. Here $E$ is a compact set, but $f(z)$ is not holomorphic. So it appears that we need some form of the converse of the theorem to prove that such a sequence cannot exist. Also I do not understand if the two parts will lead to different answers and how so.

Answer 1

Mergelyan's theorem applies to sets even without interior! In this case it only requires that the function be continuous. It is truly a powerful result.

As for the second case, what does the Cauchy integral formula tell you about a sequence of polynomials converging uniformly to some function on the unit cicle?

Answer 2

Hint: when $|z|=1$, $\overline{z} = 1/z$.

Mergelyan's theorem (if you want to use polynomials) requires ${\mathbb C} \backslash E$ to be connected.