I have to find a harmonic function F on the unit disk $D_1(0)$, continuous on the closed disk $\{z | \; \vert z \vert \le 1\}$, such that for $ t\in \mathbb{R}$:
$$F(e^{it})= \sum_{k\in \mathbb{Z}} \frac{e^{(ikt)}}{k^2-3}$$
The function we find can be written as a serie, but I only understand that if $z \in \partial D_1(0)$, $$F(z)= \sum_{k\in \mathbb{Z}} \frac{z^k}{k^2-3}$$
This function is also harmonic on $\partial D_1(0)$ but diverges inside the disk. So it is not continuous inside. I don't know where to find (maybe with Fourier series), so if you have an idea, don't hesitate!
A function $f$ harmonic on the unit disk, such that $f_{\partial D}=F$ can be found (if $F$ satisfies certain regularity conditions, namely Dirichlet' conditions) with the expression I wrote in the comments (here $\hat{F}(n)$ is the n-th fourier coefficient of $F$)
$f(\rho,\theta)=\sum_{n\in\mathbb{Z}}\hat{F}(n)e^{in\theta}\rho^{|n|}$.
In fact $f$ is armonic (it is enough to do this work for a generic term in the sum, and it could be easier if you would use polar coordinates), and, since $F$ satisfies D.C., then $f(1,\theta)=F(\theta)$
From here, finding $\hat{F}$ for your specifical $F$ is easy, since it has been given already as a Fourier series.
If you are interested on some generalizations, for example working on a sphere instead that on a circumference, have a look at spherical harmonics and at Peter-Weyl theorem