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 $b$?
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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).
subrosar
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Thank you. So given $a$ and $b$ do you think we can characterize a class of functions (with possible finite parameters) i.e. $(f(x;\alpha_1,\dots,\alpha_n),\ g(y;\beta_1,\dots,\beta_m))$? – Aug 16 '19 at 01:39