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I hope somebody can help me...

Studying mean field theory in a passage it was necessary to calcolate the inverse of this operator defined on Z^2:

$A(I,K)=-J\sum_e \delta(I,K-e)+1/(\beta)*\delta(I,K)$

where I,K pass all ZxZ and the sum on $e$ is a sum on the for basis vectors e_1,e_2,...,e_4. $\delta(A,B)$ is the usual delta function. $J$ and $\beta$ are constants.

well my book tries to compute $A'(q,p)$ as the discrete time fourier transform of $A(I,K)$... then finds a certain function $g$ which respects this equation $A'(q,p)*g(q,p)=\delta(q-p)$ and anti-transforms it, pretending thus to find an integral representation of the inverse matrix....

unluckily I don't see why this passage is true... does anybody can help me?