Problem statement: Given a proper closed arc of the unit circle, is there a sequence of polynomials that approximates $\bar{z}$ on the arc?
My thoughts:
- Well $\bar{z} = 1/z$ on the arc...But the definition of being holomorphic on a compact set is being holomorphic on an open set that contains it? And hence I cannot directly use Runge's theorem.
- Along this line of thoughts Complex analysis - $\bar{z}$ cannot be uniformly approximated by polynomials in $z$ on the closed unit disc in $\mathbb{C}$., since $|z||p(z)-\bar{z}|=|zp(z) -1| <\epsilon$, this problem is converted into whether we can use polynomials without constant terms to approximate a constant function...That is still not in the range of Runge's theorem....