I understand the continuous version of the Fourier Slice-Projection theorem, which says that given a (nice enough) function $f:\mathbb{R}^2\to\mathbb{C}$ the following operations give the same result:
Evaluate f on a line through the origin and perform a 1-d Fourier transform of the thus obtained 1-d function
Perform a 2-d Fourier transform of $f$ and project (integrate) it along the direction orthogonal to the line used in (1).
Question: What is the analogue of the Slice-Projection theorem for Fourier series?
In more detail, suppose $f:\mathbb{R}^2\to\mathbb{C}$ is a smooth function that is periodic in the sense that there exist vectors $u,v\in\mathbb{R}^2$ such that for all $m,n\in\mathbb{Z}$ and all $x\in\mathbb{R}^2$ we have $f(x)=f(x+mu+nv)$. Such a function has a Fourier series representation and there should be a relationship between projections of Fourier coefficients and slices of the function.
Assume for the moment that $u$ and $v$ are orthogonal. It is easy clear that the slice projection theorem is still true for directions aligned with the coordinate axes. However, it is not obvious to me what a projection of Fourier coefficients should be for an arbitrary direction. I would be happy with results that only apply to certain directions in which the slice is made (the relevant ones will likely be such where the 1-dimensional slice of the function is still periodic). Unfortunately I haven't been able to find the solution to this problem in the literature. Thank you very much in advance for any thoughts or suggestions!
Note: I essentially asked this question before but received no response despite a bounty. Now I came to realize that the question was phrased very poorly and needs to be reformulated completely. I present it now in two dimensions instead of three and without reference to numerical linear algebra. Since this changes everything about the question I decided to make it a new one