Can we solve for $a$ in $b(x) =\int_{-\infty}^\infty b(s)a(x,s)ds$

Question

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 $a$?

Answer 1

The solution for your integral equation is the Dirac Delta function, that is $$a(x,s) = \delta(s-x).$$ It is known as the sifting propert of the dirac delta function.

Answer 2

Uniqueness is not necessarily guaranteed. For example, when definable, take $a(x,s)=\frac{b(x)}{b(s)}f(s)$ where $f$ is any probability density, that is $\int_{-\infty}^\infty f(s) ds=1$.