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Hi

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 (hereRis 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

[tex]

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

[/tex]

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

[tex]

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

[/tex]

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

[tex]

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

[/tex]

it is quite obvious that all terms are there, i.e. it is not only thek=0term that survives. That is a paradox I cannot explain. Can you guys shed some light on this?

Best,

Niles.

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# Homework Help: Lattice discrete Fourier transforms

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