# Projection-Slice Theorem for Fourier series

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

• It's possible that you got no response last time because there is no such analog, or at least not that anyone is aware of... Jul 20, 2018 at 16:04
• I do not know a reference, but it is not toooo hard to prove something in the case that the "slice" is at a rational angle, so that the restriction is periodic of some period. That version makes sense, also, in the context of what is sometimes called a "trace theorem", about restrictions to nicely-imbedded submanifolds of functions in some Sobolev class. It is unclear to me what correct assertion(s) could be made in the non-periodic case, since spaces of "almost periodic functions" are subtler than periodic, in the first place, and, second, I don't know what one might be wanting from them... Jul 20, 2018 at 19:19
• Thank you very much for your comment! Can you maybe give a hint at what the "something one can prove" turns out to be in the case of rational angles? That special case would be completely sufficient for me, however I can't find the correct claim. The relation between "trace theorems" and my problem turned out to be above my level in terms of distribution theory. I'm also mostly interested in finding the claim rather than rigorous proof. If the distribution theory underlying "trace theorems" will help me I'm of course willing to learn about it but currently fail to see the relation. Jul 23, 2018 at 8:04