For all $x \in \mathbb{R}$, $b(x) =\int_{-\infty}^\infty b(s)a(x,s)ds.$ If it helps, we can assume that $a, b$ are continuous, nonnegative, and $\int_{-\infty}^\infty$ of $a$ or $b$ are both bounded.
Two questions: (1) Is $a$ unique? and (2) to what extent can we solve for $a$?