I've aware that a discrete approximation of an integral can be made using matrix multiplication. I seek to find the continuous analog of the matrix multiplication $(GD^{-1})(GD^{-1})^T\vec{x}$. Where $D \in \mathbb{R}^{n\times n}$ and $G \in \mathbb{R}^{m\times n}$.
\begin{array}{|c|c|} \hline Discrete & Continuous \\ \hline \vec{p}=[D]\vec{w} & p(x,y) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} d(x-x_0,y-y_0) w(x_0,y_0) dx_0 dy_0\\ \hline \vec{N}=[G]\vec{w} & N(\psi_1,\psi_2) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(x_0,y_0,\psi_1,\psi_2) w(x_0,y_0) dx_0 dy_0\\ \hline (GD^{-1})(GD^{-1})^T\vec{x} & ??? \\ \hline \end{array}
If it helps, I know that $D$ is circulant (which can also be seen because the continuous analog of $\vec{p}=[D]\vec{w}$ is a convolution). As a result, I can use Fourier transforms $F$.
\begin{equation} F[p(x,y)] = F[d(x,y)]\times F[w(x,y)] \end{equation}
\begin{equation} w(x,y) = F^{-1} \Big[ \frac{F[p(x,y)]}{F[d(x,y)]}\Big] \end{equation}