If the elegant Weierstrass approximation theorem true for complex functions? Meaning, if $\displaystyle f(z)$ be a continous function on a compact set $\displaystyle S$ (non-empty) then for any $\displaystyle \epsilon > 0$ there exists a polynomial $\displaystyle g(z)$ such that $\displaystyle |f(z)-g(z)| < \epsilon$.
If the elegant Weierstrass approximation theorem true for complex functions? Meaning, if $\displaystyle f(z)$ be a continous function on a compact set $\displaystyle S$ (non-empty) then for any $\displaystyle \epsilon > 0$ there exists a polynomial $\displaystyle g(z)$ such that $\displaystyle |f(z)-g(z)| < \epsilon$.
That depends on what you want to take for the set S. If S is a subset of the complex plane then the result is false. For example, on the unit circle the complex conjugate function $\displaystyle f(z) = \bar{z}$ is not a uniform limit of polynomials (because a uniform limit of analytic functions has to be analytic).