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


$\begin{eqnarray}
 p(x,y) &=& c\_0+c\_1x+c\_2y+c\_3xy+c\_4x^2+c\_5y^2+c\_6xy^2 \\
 G(x,y) &=& c\_k\cdot\exp\left(\frac{-(x^2+y^2)}{2\sigma^2}\right) \\ 
 \square\_a(x,y) &=& \mathbf{1}\_{[-a,a]\times[-a,a]}(x,y)
\end{eqnarray}$


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 a better way of performing the deconvolution?


Thank you very much


EDIT: Maple says its nothing but a polynomial


$$-\frac{1}{8}\frac{(c\_4+c\_5)\cdot \sigma^2-c\_0}{\pi\cdot c\_g\cdot \sigma^2\cdot a^2}$$



How is the deconvolution of a fat gaussian from a polynomial derived?
In a lax enough environment, the Fourier transform of $x^n$ is the distributional derivative of the $\delta$ distribution. The relevant behavior of the resulting distribution is: $$\delta^{(n)}[\phi]=(-1)^{n}\cdot \phi^{(n)}(0).$$


From this the result quickly follows.


