I was just wondering about what the Fourier transform of a parallelepiped filter function would be in 3D. The function would look something like
$f(\mathbf{r}) = \left\{ \begin{array}{ll} 1 & \text{if $\mathbf{r}$ is inside the parallelepiped} \\ 0 & \text{otherwise} \\ \end{array} \right. $
The one-dimensional analogue would be a top-hat/boxcar function whose Fourier transform pair is a sinc function. In 3D, the Fourier transform of a cubic filter function would then be a 3D sinc function (separable product of 1D sinc functions for each Cartesian direction if I'm not mistaken). Hence, I'm guessing that the Fourier transform of a parallelepiped filter function would look like a sinc function sheared in the shape of the reciprocal lattice defined by the real-space parallelepiped, but I'm not sure how I'd go about showing this properly.
Another thought I had was about whether there exists some kind of Fourier transform identity that tells us how Fourier transforms change when the input space undergoes some kind of linear transformation, i.e.$\mathcal{FT}\{f(A \mathbf{r})\} = ?? $ where $A$ is a linear transformation. If it exists, it seems that it would be easy to go from a cubic filter to a parallelepiped-shaped one.
Thank you!
Every parallelepiped $P = TW$ is the image of the unit cube $W$ under a linear transform $T$. Just define the linear transform by $T e_j = \tilde{e}_j$ for $\{e_j \vert \, j \in \{1, \dots, 3\} \}$ the standard basis and $\{\tilde{e}_j \vert \, j \in \{1, \dots, 3\} \}$ the vectors generating the parallelepiped.
Therefore $f = \mathbb{1}_{P} = \mathbb{1}_W \circ T^{-1} $.
The fourier transform $\mathcal{F}$ obeys $$ \mathcal{F}f(\xi) = \mathcal{F} (\mathbb{1}_W \circ T^{-1}) (\xi) = \vert \det T \vert \, (\mathcal{F}\mathbb{1}_W)(T^t \xi) = \mathrm{Vol}(P) \, (\mathcal{F}\mathbb{1}_W)(T^t \xi). $$
So turning to the fourier transform of the unit cube indicator function, using Fubini/Tonelli, we get $$ \mathcal{F}(\mathbb{1}_W)(\xi) = \mathcal{F}(\mathbb{1}_{[0,1]})(\xi_1) \, \mathcal{F}(\mathbb{1}_{[0,1]})(\xi_2) \, \mathcal{F}(\mathbb{1}_{[0,1]})(\xi_3) $$ and the fourier transform of the indicator function of an interval is just a (modulated) sinc function as you already mentioned.