Let let $i,l=1,...,N$, let $a_{i}$ and $b_{i,l}$ be some positive constants with $\sum_i b_{i,l} = \sum_l b_{i,l} = 1$ for all $i,l$. Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$. Consider the following system of equations in unknowns $x_{i},y_{i}$: $$a_{l}=\sum_{i} b_{i,l} f(x_i,y_l),l=1,...,N$$ $$a_i=\sum_{l} b_{i,l} f(x_i,y_l),i=1,...,N.$$
I am wondering if this system is well known, or studied in some branch of mathematics.