For $\phi$ $\in$ $L_2$($R^n$) , h($\omega$)= $\sum_{k\in Z^n} \vert\hat\phi(\omega+2\pi k)\vert^2$ $\in$ $L_2([-\pi,\pi]^n)$ is called the auto correlation function. prove that if integer shifts of $\phi$ are an orthonormal system then, $h(\omega)$=1
I have tried using the relation $\widehat{\phi(x-k)}$($\omega$)=$e^{-ik\omega}\hat\phi(\omega)$ and use the fact that
<$\widehat{\phi(x-l)},\widehat{\phi(x-m)}$>=$(2\pi)^n<\phi(x-l),\phi(x-m)$> but i seem to have some variables problems.
I have also tried to use parseval theorem but i am having troubles with the fact that there is a sum and not an integration over $\hat\phi$.
what am i missing?