I'm trying to better understand how a function on the sphere $\mathbb{S}^2$ can be decomposed in terms of spherical harmonics. In particular I have found this notation, given that $f: \mathbb{S}^2 \rightarrow \mathbb{R}$, then we can write:
$$f(\textbf{x}) = \sum_{l=0}^\infty \sum_{m=-l}^la_m^lY_m^l(\textbf{x}) \, ,$$
where the family of functions $Y_m^l : \mathbb{S}^2 \rightarrow \mathbb{R}$ form a basis for $\mathbb{L}^2(\mathbb{S}^2)$. I'm more familiar with the notion of Fourier Transform, where we have a basis of complex exponentials $\{e^{ilx}\}_{l \in \mathbb{Z}}$ that allow to describe any periodic function as weighted sum of sines/cosines in the following way:
$$f(x) = \sum_{n \in \mathbb{Z}}f_n e^{inx}$$
with $f_n$ being the Fourier coefficients. So I'm trying to relate this second statement to the first one, in particular I'm a bit confused about why do we have two sum operators on the first one, and the precise role of the indexes $l$ and $m$.