Functional Equality

Question

If I have the following equations: $$a(r)=\int_0^\infty s\ f(rs)\ g(s(1-r))\ ds\\b(s)=\int_0^1s\ f(rs)\ g(s(1-r))\ dr$$ Where $f,\ g>0$, $s\in (0,\infty)$ and $r\in (0,1).$ Is it possible to write $f$ and $g$ in terms of $a$ and $b$?

Answer 1

Before talking about whether $f$ and $g$ can be expressed in terms of $a$ and $b$ it is worth trying to ask whether $f$ and $g$ are even uniquely determined by $a$ and $b$. The answer is no.

Given $a$ and $b$ if some pair of functions $(f,g)$ satisfy the given integral equations then for any non-zero constant $c$ the pair $(cf,g/c) will also satisfy those equations.

So $f$ and $g$ are not unique. It is possible that they are unique up to a scale factor (although I doubt it).