Polynomials dense in $C(S^1)$?

Question

Consider $C(S^1)$ the continuous functions from $S^1$ to $\Bbb C$, equipped with sup norm. Can any function $S^1\to \Bbb C$ be approximated by polynomials?

Answer 1

Assuming you mean polynomials in $\Bbb C[z]$ restricted to $S^1\subseteq \Bbb C$, the answer is no. If $p:\Bbb C\to \Bbb C$ is a polynomial, then $\int_0^{2\pi} ie^{i\theta}p(e^{i\theta})\,d\theta=0$. Thus, if $p_n\to f$ uniformly on $S^1$, then, $$\int_0^{2\pi} ie^{i\theta}f(e^{i\theta})\,d\theta=\lim_{n\to\infty}\int_0^{2\pi}ie^{i\theta}p_n(e^{i\theta})\,d\theta=0$$

However, the continuous map $z\mapsto\overline z$ does not satisfy this identity.