2D discrete Fourier transform

Hi guys

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

$\displaystyle f\mathbf{k}) = \sum_{\mathbf{R} } e^{i \mathbf{k} \cdot \mathbf{R} } g(\mathbf{R})

$

where **k** is a vector. Now, since each site contains the number 1, the system is homogeneous, and from the inverse Fourier transform,

$\displaystyle g(\mathbf R) = \sum_{\mathbf k} e^{-i\mathbf k\mathbf R} f(\mathbf k)$,

we see that only the **k**=**0**-term can survive, since g(**R**) is constant. But by performing the sum

$\displaystyle f(\mathbf{k}) = \sum_{\mathbf{R} } e^{i \mathbf{k} \cdot \mathbf{R} } g(\mathbf{R})$,

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?

Best,

Niles.