A corollary to Stone-Weierstrass theorem

Question

Can i get the answer to the following problem. I am having a proof, i feel there is something wrong here..Can you please point out!

Let $D\subset \mathbb C$ be a simply connected domain, and $\gamma: [0,1]\to D$, be a smooth embedding. Given a continuous one form $\phi$ along $\gamma$ and $\epsilon >0$, Does there exists a holomorphic function $h$ on some open neighborhood $U$ of $\gamma$, $U\subset D$ such that $|dh-\phi|<\epsilon$.

Suggested Proof:

Without loss of generality we can assume that $0\notin D$. We can write $\phi= \phi_1 d\zeta$, with $\phi_1$ a continuous function on $\gamma$. We can uniformly approximate $\phi_1$ by Laurent polynomials of the form $\psi_1(\zeta)= \sum_{-k}^k a_j\zeta^k$. As $0\notin D$, we have $\psi_1(\zeta)$ analytic on some possibly small simply connected subdomain of $D$ which we will denote by $D$ itself.

We know that if D is a simply connected domain and $\psi_1$ is analytic in D, then $\psi_1$ has an antiderivative at all points of D. Hence take $h(z)= \int \psi_1(\zeta)$ which will be our required holomorphic function.

Answer 1

In your case we can find a holomorphic function on the plane that uniformly approximates the given continuous function .It is a consequence of the following .Suppose K is a compact measure zero subset of the plane whose complement in the plane is connected then every continuous function on K can be uniformly approximated by entire functions. Hartogs-Rosenthal says any continuous function on K can be uniformly approximated by functions holomorphic in a neighbourhood of K.By Runge's theorem, functions holomorphic in a neighbourhood of K can be uniformly approximated on K by entire functions. In your case since gamma is an arc its complement is connected .Since it is smooth it has measure zero .