I am struggling with the following proof.
Prove that if $\psi\in L^2(\mathbb{R})$ then $\{\psi(t-n)\}_{n=-\infty}^{\infty}$ is an orthonormal set if and only if $\Psi(\xi)=\sum_{n=-\infty}^{\infty}|\hat\psi(\xi+n)|^2=1$ almost everywhere.
I don't know what is considered standard notations so for the sake of clarity, $L^2(\mathbb{R})$ is a Lebesgue space and $\hat\psi$ is the Fourier transform of $\psi$.
I am just very confused as to how I even start here. I tried just doing inner products but I couldn't get anywhere.
Any help is appreciated.