During my studies I came across the following integral $$\oint_S\mathrm{d}S\,\mathrm{e}^{-\mathrm{i}\vec{x}\cdot\vec{k}},$$ where $S$ is a 2D surface in 3D space, and noticed that it almost looks like a Fourier transform of the surface itself. I was wondering (and it would be very helpful) if there is a way to express this in terms of some series in $\vec{k}$, or if there are theorems related to FTs of surface elements? The only resources I could find dealt with surface measures of spheres.
Edit: Rethinking the question, since I implicitly assume that my surface is closed, what I am actually doing is not a Fourier transform per se, but rather a Fourier series of the "surface itself". In this context, I'm wondering if there is anything to say about the expansion coefficients of the series in $\vec{k}$?
Perhaps to give a little more context to what I'm thinking about, let's say we can expand $S$ using spherical harmonics as $$S(\theta,\phi)=r_0\sum_{\ell=0}^\infty\sum_{m=-\ell}^\ell s_{\ell m}Y_{\ell m}(\theta,\phi),$$ where $s_{\ell m}$ are constants. Then the integral can be written as $$\int_0^{2\pi}\mathrm{d}\phi\int_0^{\pi}\mathrm{d}\phi\,\lVert\partial_\theta S\times\partial_\phi S\rVert\mathrm{e}^{-\mathrm{i}S(\theta,\phi)\hat{r}\cdot\vec{k}},$$ where $\hat{r}$ is the radial unit vector of $(\theta,\phi)$. Writing it like this then suggests that it is (hopefully?) possible to expand the integral in some way like $$\oint_S\mathrm{d}S\,\mathrm{e}^{-\mathrm{i}\vec{x}\cdot\vec{k}}\propto\sum_{\vec{n}\in\mathbb{Z}^3}c_{\vec{n}}(s_{\ell m})\mathrm{e}^{\mathrm{i}\vec{n}\cdot\vec{k}}.$$ My hope is there are already some results or something similar.