From Orthogonality to DTFT I have met this question, but cannot prove it using DTFT definition.
Given: $g$ is a discrete sequence filter and:
$$ g \in l^2(Z)$$
$$\langle g_n, g_{n-2k} \rangle = \delta_{k}$$
Prove:
$$|G(e^{j \omega})|^2 + |G(e^{j(\omega + \pi)})|^2 = 2$$

Could you guys give me some direction on this one?
Thank a lot
Here a few hints to get you started:


Define a sequence $x[k]=\langle g_n, g_{n-k} \rangle$ and write it as a convolution. From that it should be easy to find its DTFT $X(e^{j\omega})$ in terms of $G(e^{j\omega})$.
Next note that the given sequence is just a downsampled version of $x[k]$, so its spectrum is an aliased version of $X(e^{j\omega})$. That's how you end up with those two terms on the left-hand side of the frequency-domain representation.