Here is an unknown luminosity function $f(x,y)$ and its integration results: $\begin{align*} p_{i,j} &= \frac{1}{\Delta_{i,j}}\iint\limits_{D_{i,j}} \! f(x,y) \, dx \, dy,\\ \Delta_{i,j} &= \iint\limits_{D_{i,j}} \!dx\,dy\;. \end{align*}$ Let's consider the following transformations: $\begin{align*} F(x,y,r,\sigma)&=\frac{1}{2 \pi \sigma^2}\int\limits_{x-r}^{x+r} \int\limits_{y-r}^{x+r} \! f(u,v) \,e^{-\frac{(u-x)^2+(v-y)^2}{2\sigma^2}} \, du \, dv\;,\\ q_{i,j}(r,\sigma)&=\frac{1}{\Delta_{i,j}}\iint\limits_{D_{i,j}} \! F(x,y,r,\sigma) \, dx \, dy\;. \end{align*}$ Is there a functional relationship between:
- $p_{i,j}$ and $q_{i,j}(r,\sigma)$, where $r \in [0,+\infty)$;
- $p_{i,j}$ and $q_{i,j}(\infty,\sigma)$?