In this question, a “(closed oriented plane) **curve**” $\Gamma$ will mean a continuous map $f \colon \mathbb{U} \to \mathbb{C}$ where $\mathbb{U} := \{z\in\mathbb{C} : |z|=1\}$ is the unit circle, modulo right-composition by orientation-preserving homeomorphisms $\varphi\colon\mathbb{U}\to\mathbb{U}$. Any such $f$ (i.e. any of the $f\circ\varphi$ defining the curve) will be called a “parametrization” of the curve $\Gamma$.

I'm willing to add any reasonable regularity conditions on $\Gamma$ if they help in answering the question, e.g., piecewise $C^1$ or even $C^\infty$.

Say that a curve $\Gamma$ is **harmonically parametrizable** iff it admits a parametrization $f \colon \mathbb{U} \to \mathbb{C}$ (see above) such that the Fourier coefficients $(c_k(f))_{k\in\mathbb{Z}}$ of $f$ are zero for all $k<0$: this then allows us to see $f$ as the restriction to $\mathbb{U} = \partial\Delta$ of a continuous function $F$ on the closed unit disk $\overline{\Delta}$ that is holomorphic on the open unit disk $\Delta$ (namely, $F$ is the Poisson integral of $f$, or equivalently $F(z) = \sum_{k=0}^{+\infty} c_k z^k$; see also this question).

More generally, if $k_0\in\mathbb{Z}$ say that $\Gamma$ is **$k_0$-harmonically parametrizable** iff it admits a parametrization $f \colon \mathbb{U} \to \mathbb{C}$ that $c_k(f)=0$ for all $k<k_0$. (So “harmonically parametrizable” means “$0$-harmonically parametrizable”, and the larger $k_0$ the stronger the condition.)

For example, if $\Gamma$ is a positively-oriented, sufficiently regular (I think “rectifiable” suffices), Jordan curve, then the Riemann conformal mapping theorem guarantees that we can find $F \colon \Delta \to V$ bijective and conformal, where $V$ is the bounded component of the complement of $\Gamma$, extending continuously to $\partial\Delta = \mathbb{U}$ (assuming $\Gamma$ is sufficiently regular), which shows that $\Gamma$ is harmonically parametrizable.

So, two related questions:

Can we find a necessary and sufficient condition on a curve $\Gamma$ (again, assumed sufficiently regular if this is useful) for it to be harmonically parametrizable?

What about $k_0$-harmonically parametrizable for $k_0<0$? Can we find, at least, a reasonable sufficient condition analogous to the “Jordan curve” condition for being harmonically parametrizable?

(For $k_0>0$, being $k_0$-harmonically parametrizable seems to say something about the moments of $\Gamma$ for the harmonic measure vanishing, and this is probably less interesting.)

**PS:** This question seems related to a kind of converse to the one I'm asking (I want to know if there exists a parametrization change which becomes harmonic, that other question is about what happens upon such a change).