The inner product of the two-dimensional sequences $f(x,y)$ and $g(x,y)$ is equal to the inner product of their Fourier transforms, that is:
$\sum_{x=-\infty}^{\infty}\sum_{y=-\infty}^{\infty}f(x,y)g^*(x,y)=\dfrac{1}{4\pi^2}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}F(w_x,w_y)G^*(w_x,w_y)\,dw_x\,dw_y.$
I am trying use a Fourier transform inverse and follow re-arranged the integrals and use the Dirac function. But I don't know Why the integrals have limits $(-\pi,\pi)$.