Let $f\in L^2(\mathbb R)$. Prove the system $\{f(t-n)\}_{n\in\mathbb{Z}}$ is orthonormal if and only if $$\sum_{k\in\mathbb{Z}}|\hat{f}(\omega+2\pi k)|^2\equiv 1$$
I have no clue how to prove both directions. I think that somehow Parsevall's equality has to be applied but I'm not sure how. I'd be glad for any hint.
Take any advanced wavelet book, this is a fundamental relation for the examination of affine systems.
As you surely know, $$ c_n=\int_{\Bbb R}f(t)\overline{f(t-n)}\,dt =\frac1{2\pi}\int_{\Bbb R} |\hat f(ω)|^2·e^{-inω}\,dω =\frac1{2\pi}\int_{-\pi}^\pi\sum_{k\in\Bbb Z}|\hat f(ω+k·2\pi)|^2·e^{-inω}\,dω $$ Now the first and last term are about the Fourier series of the $2\pi$ periodic function $$ g(ω)=|\hat f(ω+k·2\pi)|^2=\sum_{n\in\Bbb Z}c_ne^{inω}. $$ Orthonormality is equivalent to $c_n=\delta_{0,n}$ which is equivalent to $g\equiv 1$.