Say I have a 5x5 lattice, where each entry (or we can call it site) contains the number 1. Now, on the lattice we have a function g(R), which is equal to the number on the site. In this case g(R)=1 for all sites (here R is a vector from the point (3,3), which denotes the site we are talking about).
Now I wish to Fourier transform the function g, and I use the lattice discrete FT
where k is a vector. Now, since each site contains the number 1, the system is homogeneous, and from the inverse Fourier transform,
we see that only the k=0-term can survive, since g(R) is constant. But by performing the sum
it is quite obvious that all terms are there, i.e. it is not only the k=0 term that survives. That is a paradox I cannot explain. Can you guys shed some light on this?