I'm stuck on what should by all intents be a simple enough question. Suppose $n$ is an integer not divisible by 2 or 3 and let $f : \mathbb{Z}_n \to \mathbb{C}$ be a function and let $\hat{f}$ be it's discrete fourier transform, defined on $\mathbb{Z}_n$ as $\hat{f}(r) = \mathbb{E}_x(f(x)\omega^{-rx})$ where $\omega = e^{2\pi i/n}$.
For background info we can furthermore define the inner products $\left<f,g\right> = (\mathbb{E}_x f(x)\overline{g(x)})$ for functions and $\left<\hat{f},\hat{g}\right> = (\sum_r \hat{f}(x)\overline{\hat{g}(x)})$ for Fourier transforms (we do it in this way to avoid having to deal with normalising factors in Fourier Inversion etc).
Then it is easy to show that Fourier inversion holds in the form $f(x) = \sum\limits_r \hat{f}(r) \omega^{rx}$ and also that Parseval holds in the form $\left<f,g\right> = \left< \hat{f},\hat{g}\right>$. Furthermore convolutions behave as expected, i.e. $\widehat{f * g} = \hat{f}\hat{g}$, where the convolution $f*g(x) = \mathbb{E}_y (f(y)g(x-y))$. This should be all the background info that is necessary.
Using this it should be possible to work out an expression in terms of $\hat{f}$ for the quantity $\mathbb{E}_{x,d} (f(x)f(x+d)f(x+2d)f(x+3d))$, however I am not able to get very far as despite whatever I do I am not able to get to a position where it is possible to apply Parseval.
Any help is much appreciated!