Definition of Fourier Transform on a Lattice

I am reading a book(EDIT: the book is Czyholls theoretical condensed matter physics, though i am not sure if there is an english version) where for periodic functions $f(x_l+L)=f(x_l)$ the Fourier transform is defined as:

$$f(x_n)={1\over N} \sum \limits_l e^{iq_lx_n} f(q_l)$$ and the backtransform $$f(q_l)=\sum \limits_n e^{-iq_lx_n} f(x_n)$$

I understand how you can prove that the Bravias lattice function is the result of a Fourier transformation of the real space lattice function. But in these cases the usual integral definition of the Fourier transformation was used.

Is this just because the function is defined only on the lattice so $f(x_n)$? Is there any material getting into all the details of this discrete fourier transformation, but in a condensed matter context?

• Perhaps have a look at Pontryagin duality and the Fourier transform. The Fourier transform works on far more general domains of function than $\mathrm{R}^n$, but there are not really any "details" to it (it's just replacing the integral with appropriate sums in the cases you seem to be interested in), so I'm unsure what you're asking. – ACuriousMind May 16 '15 at 12:05
• – Qmechanic May 16 '15 at 12:20
• @ACuriousMind I am halfway satisfiyed with dourier theory because i view it as a projction on the eigenvectors of the differential operator, witch i know to ber hermitian. So i would like to know how i can have some analogus theory in this case, but i will have a closer look at thi de.wikipedia.org/wiki/… i just found. I was just asking in case someone knew a condensed matter or something like that to actually explain why this makes sense. – pindakaas May 16 '15 at 12:38
• But actually just knowing that discrete fourier transformation is a an independent theory that is used in condensed matter theory is already helpfull. I don't think i have seen this word in any book on condensed matter theory to this day... – pindakaas May 16 '15 at 12:45
• What if the system is finite? For example, consider a two leg ladder, does it make sense to perform the Fourier transform along the rung? – ZJX Jan 11 at 0:10