Inverse of operator with kernel $K(x,y)=(1-xy)^{-1}$ on the unit interval Suppose $$f(x)=\int_0^1 \frac{g(y)}{1-xy}dy.$$ Is there a $G(x,y)$ such that 
$$g(y)=\int G(x,y)f(x)dx?$$ (with some integration contour)

So we have an operator $T$ bringing $g(x)$ into $(Tg)(x)=\int_{0}^{1}\frac{g(y)}{1-xy}\,dy$ and we want an explicit representation for $T^{-1}$, fine. We may notice that $[x^n](Tg)(x)=\int_{0}^{1}y^n g(y)\,dy$, so the reconstruction of $g$ from $Tg$ is equivalent to solving the moment problem. Under suitable regularity assumptions,  it can be done through the Laplace or Mellin transform.
