We have a 2D order-2 polynomial, a Gaussian and a 'box' indicator function. Let:

$\begin{eqnarray} p(x,y) &=& c_0+c_1x+c_2y+c_3xy+c_4x^2y \\ G(x,y) &=& \exp\left(\frac{-(x^2+y^2)}{2\sigma^2}\right) \\ \square_a(x,y) &=& \mathbf{1}_{[-a,a]\times[-a,a]}(x,y) \end{eqnarray}$

all functions of reals, all real constants, and $a,\sigma>0$.

Let $\otimes$ denote convolution, and $\mathcal{F\{\dots\}}$ denote the Fourier transform.

Is there a "closed form" for the deconvolution of $\square_a\otimes G$ from $p$ (when $\mathcal{F}\{\square_a\otimes G\}$ is away from 0)? In other words, can the following expression be significantly reduced?

$$\mathcal{F}^{-1}\left\lbrace\frac{\mathcal{F}\{p(x,y)\}}{\mathcal{F}\{\square_a\}\cdot \mathcal{F}\{G\}}\right\rbrace $$

I am trying to start computing this in Maple.

This looks like the inverse Fourier transform of a bunch of derivatives delta functions divided by a sinc scaled by a Gaussian. Is there any reason to believe that this has a nice closed form? If so, is there a simpler way of finding its deconvolution?

Thank you very much