I am wondering if I can find a function $ Q(x,\rho)$ if I am given the following equation:
$B(\alpha,\rho) = \iint_{\mathfrak{R}} \delta[\alpha - Q(x,y,\rho)]\, \text{d}x \text{d}y$
Here, $\mathfrak{R}$ is some region and we are integrating over the Cartesian $(x,y)$ coordinates. I am assuming I know the value of $B(\alpha,\rho)$. Is it possible to find $ Q(x,y,\rho)$?
I think we we integrate both sides of the equation, after multiplying by $\alpha$, over all $\alpha$, then we can arrive at the following equation:
$\overline{B}(\rho) = \iint_{\mathfrak{R}} Q(x,y,\rho)\, \text{d}x \text{d}y$
Here $\overline{B}(\rho) = \int \alpha B (\alpha,\rho) \, \text{d} \alpha$. So now we have some type of integral equation. Is it possible to solve this for $ Q(x,y,\rho)$ somehow?