Consider the transforms
$$ A(f(x),g(x)) = h(x) = 1 + \int_0^x f(x - t) g(t) dt $$
$$ B(h(x),g(x)) = f(x) $$
$$ C(h(x),f(x)) = g(x) $$
Where the functions on the LHS are given.
Notice $B,C$ are the inverses of $A$.
How to compute $B,C$ ?? Is there an integral representation for $B$ or $C$ ?
How to find
$$ f(x) = h(x) $$
For a given $g(x) $ ?
How to find
$$ g(x) = h(x) $$
For a given $f(x) $ ?
It reminds me of Laplace transforms and convolution ... not sure If that helps.