Can one uniformly approximate a function 'similar' to identity on $S^1$ with complex polynomials? I mean a function like:
$f(z)=z \cdot (1+h \cdot \sin(m\cdot Arg(z)))$, for $|h| < 1,\ m \in \mathbb{N}$.
What I actually have is a function (actually I have a sequence of them but that is another story) like the following one (or a smooth version of it):
$f(z)=z \cdot (1+ h \cdot wave(Arg(z)))$, where
$wave(\phi)=[\phi^{-1} \in [\pi,k\pi]] \cdot \sin(\frac{1}{\phi})$, $[p]$ is $1$ if $p$ is true and $0$ otherwise and $k \in \mathbb{N_+}$.
Can anyone tell me if there are any theorems saying that this approximation can be done or not? It looks similarly to Fourier series. What actually needs to be done is approximating $wave(Arg(z))$. However I need $z^{-n}$ to use Fourier series theory and I have only polynomials. $z^{-n}$ can not be approximated on the whole $S^1$ with polynomials.
If I was able to extend the function to the whole 'unit+$\varepsilon$' disk in a holomorphic way I would be able to use Taylor series for this function, but I don't know if such an extending can be done, since the intuitive way with going orthogonally from the $f(S^1)$ when one goes orthogonally from the $S^1$ leads nowhere.
My question comes from a bigger problem and what I actually need to solve it is: make such an approximation $w$ of $f$, that:
- $w(S^1) \subseteq{} \lbrace z \in \mathbb{C}: dist(z,f(S^1))< \delta \rbrace$
- $|z-w(z)| < \varepsilon(h)$
where $\delta$ is some unknown constant depending on $f$. We may choose as $\varepsilon$ any function that is is continuous at zero and $\varepsilon(0)=0$.
This one looks more complicated but is weaker than the previous one.
If anyone could help me with that I would be really grateful.