$X:=\{z \in \mathbb C : |z|\le 1\}$ ; is $\{p(z,\bar z) | p(x,y) \in \mathbb R[x,y] \}$ dense in $C(X , \mathbb C) $ under sup metric?

Question

Let $X:=\{z \in \mathbb C : |z|\le 1\}$ , is it true that any element in $C(X , \mathbb C)$ can be uniformly approximated by polynomials , in $z, \bar z$ , with real co-efficients ? If we wanted uniform approximation by polynomials in $z, \bar z$ , with complex co-efficients then I know it would be true by Complex version of Stone-Weierstrass theorem . But I have no idea what happens if we want real-coefficients ( I can't see whether the set $\{p(z,\bar z) | p(x,y) \in \mathbb R[x,y] \}$ is a subalgebra over $\mathbb C$ or not ) . Please help . Thanks in advance

Answer 1

To expand on the counterexample $p(z, \bar z)=iz$ given by Cameron Williams, note that when restricted to a subinterval of real line, such as $[-1,1]$, the set $\{p(z,\bar z) | p(x,y) \in \mathbb R[x,y] \}$ reduces to $\mathbb{R}[x]$. Clearly, any function one can approximate by the elements of $\mathbb{R}[x]$ must be real-valued on $[-1,1]$, and this does not include $iz$.