Determining Fourier Coefficients by inspection I'm beginning to learn about Fourier series/transforms. My teacher hopes that by now we should be able to examine a simple potential function and decompose it without having to actually do the transform.
The potential function we are given is : $$U(\vec{r}) = -4 U_0 \mathrm{cos}(\frac{2\pi x}{a})\mathrm{cos}(\frac{2\pi y}{a})$$ with which it is possible to find another function that satisfies $$U_q = \int_V \frac{d\vec{r}}{V}e^{-i\vec{q}\cdot\vec{r}} U(\vec{r})$$ where $V$ is the volume of a unit cell (or the area, or length, depending on dimensions).
It is easy to solve for $U_q$ with a table and/or trig identities. The problem is that this integration is done over the space of one unit cell and I have trouble interpreting it, as usually a Fourier Transform would be over all space.
My questions are:
1) How do you determine the coefficients by inspection? Practice, or is there a trick?
2) Why is $U_q$ defined over the space of a unit cell and not over all space?
This is very new to me and my questions might be confusing.
For the interested, this is taken from exercise 8.1 in Marder's Condensed Matter Physics 2nd Ed
 A: 
1) How do you determine the coefficients by inspection? Practice, or is there a trick?

For a contrived example like $U(\vec{r}) = -4 U_0 \mathrm{cos}(\frac{2\pi x}{a})\mathrm{cos}(\frac{2\pi y}{a})$ it's fairly simple to do, since you can just use $\mbox{cos}(2\pi x/a)=\frac{e^{2\pi i x/a}+e^{-2\pi i x/a}}{2}$ and similar for $\mbox{cos}(2\pi y/a)$, multiply them out, and use the fact that the basis for Fourier space is tensor products of plane waves, $e^{2\pi i k_1 x/a}e^{2\pi i k_2 y/a}$. 
Hopefully you're aware of this fact, but Fourier transforms are just a particular type of change of basis, quite literally in exactly the same manner as the changes of basis you've probably done in linear algebra for matrices and vectors.

2) Why is $U_q$ defined over the space of a unit cell and not over all space?

The potential you are dealing with is an approximation to the potential experienced by electrons in the environment of a periodic crystalline lattice of nuclei, such as that encountered in metals. As a result, $U(x,y)$ is periodic in both directions, with the periodicity being that of the internuclear separation $a$. Since $U$ is periodic over the cell, you only need to integrate over a single cell and then normalize the result by dividing by $V$. You could also integrate over space, but it would be pointless because it would give the same answer.
I've neatly skipped over a few delicate mathematical details, but that's intuitively the reason.
