The Weierstrass approximation theorem states that any continuous function $ f : I \rightarrow \Bbb R $ on a closed, bounded, connected subset $ I \subseteq \Bbb R $ can be uniformly approximated by polynomials.
Can any continuous function $ \phi : J \rightarrow \Bbb C $ on a closed, bounded, connected subset $ J \subseteq \Bbb C $ be uniformly approximated by polynomials?
What I mean is, for which subsets $ J \subseteq \Bbb C $ can all functions be approximated uniformly by polynomials.
This question is an example sheet question that I had (already supervised on $-$ non-examinable) but supervisor wasn't sure what the question meant exactly. There are some basic sets, such as closed, real intervals, that it clearly holds for, but others (such as closed unit ball) that it does not hold for. Is anyone able to shed any light on the answer. (Not just a few counter-examples, but some explanation as to why it does / does not hold on certain set (eg, because connected complement / similar).)
Thanks very much!
Best Answer
It depends on what is meant by "polynomial".
If only $\sum c_n z^n$, then every function that is uniformly approximable by polynomials must be holomorphic on the interior of $J$.
Although that condition is trivially satisfied if $J$ has empty interior, that doesn't mean that for such $J$ every continuous function is the uniform limit of polynomials. For example the unit circle has empty interior, but a sequence of polynomials converging uniformly on the unit circle converges uniformly on the closed unit disk by the maximum principle, and thus if $f$ is a uniform limit of polynomials on the unit circle, then there is a holomorphic function $h$ on the unit disk that extends continuously to the unit circle, with boundary values $f$. In particular, we have
$$\int_{\lvert z\rvert = 1} f(z)\cdot z^n \,dz = 0\tag{1}$$
for all $n \geqslant 0$. (And, in this case, that condition is sufficient.)
That phenomenon generalises, if $J$ disconnects the plane, that is, if $\mathbb{C}\setminus J$ has at least two connected components, then the bounded components of the complement of $J$ impose restrictive conditions on the continuous functions that are uniform limits of polynomials similar to $(1)$.
Mergelyan's theorem asserts the converse, if $J$ is a compact subset of $\mathbb{C}$ with empty interior such that $\mathbb{C}\setminus J$ is connected, then every continuous function on $J$ can be uniformly approximated by polynomials (in $z$ only).
If "polynomial" means polynomial in $z$ and $\overline{z}$, or equivalently polynomial in $\operatorname{Re} z$ and $\operatorname{Im} z$, then the Weierstraß approximation theorem holds for all compact $J$.