I found this Fourier series identity in a book on Harmonic analysis but the proof is inclear. Maybe it makes more sense using bra-ket formalism.
$$ \sum_{r, n \in \mathbb{Z}/N\mathbb{Z}} f(n) g(n+r)h(n+2r) = \sum_{a \in \mathbb{Z}/N\mathbb{Z}} \hat{f}(a)\hat{g}(-2a)\hat{h}(a)$$
$|f \rangle$ could be thought of as akind of wave function but the Hilbert space is functions $f: \mathbb{Z} \to \mathbb{C}$.
Let me try to write it out. In Fourier Series we have that $1 = \sum | a \rangle \langle a | $ so we can Fourier expand $f$:
$$ | f \rangle = \sum | a \rangle \langle a | f \rangle $$
Now what to make of this triple average? It equals a very simplicated sum over 5 indices
$$ \sum_{r, n \in \mathbb{Z}/N\mathbb{Z}} \;\;\sum_{a_1, a_2, a_3\in \mathbb{Z}/N\mathbb{Z}} \;\; \langle f|a_1 \rangle \langle a_1 | n\rangle \; \langle g|a_2 \rangle \langle a_2 | n+r\rangle \; \langle h|a_3 \rangle \langle a_3 | n+2r\rangle = \dots $$
Here, maybe we can try $\langle a|n\rangle = e^{2\pi i \, a n}$