I am looking for a reference to the claim that for any $f\in L^1(\partial \mathbb{D})$, where $\partial \mathbb{D}$ is the unit circle in $\mathbb{C}$, $f(e^{i\theta})=\displaystyle\sum_{n=-\infty}^\infty a_ne^{in\theta}$ for almost all $\theta$. I am inexperienced with both the Poisson kernel and Fourier transform, so I was hoping for a gentle explanation if these notions are required.
Thank you.
This is not true. In fact, Kolmogorov constructed (1923) an example of a $L^1$ function whose Fourier series diverges almost everywhere (later improved to everywhere divergent).
Om the other hand, if $f\in L^p$ for some $p>1$, it's a deep theorem by Carleson (the $p=2$ case) and Hunt ($p>1$) that the Fourier series of $f$ converges pointwise almost everywhere. See for example Wikipedia for more details.