2
$\begingroup$

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 $(-\pi,\pi)$.

  • 0
    Have you tried with Bessel-Parseval's inequality?2012-11-12
  • 0
    I know Parseval's theorem and Bessel inequality but Bessel-Parseval's where I will be able to found?2012-11-12
  • 1
    I meant the result which gives you an isometry between $L^2((-\pi,\pi)^2)$ and $\ell^2(\Bbb Z^2)$ (I actually don't know exactly the name). Then we polarize.2012-11-12

1 Answers 1