I found this useful1 : http://en.wikipedia.org/wiki/Fredholm_integral_equation
Using the mentioned Fourier transform formula, $ X(y) = \mathcal{F}_\omega^{-1}\left[ {\mathcal{F}_x[b(x)](\omega)\over \mathcal{F}_x[f(x,y)](\omega)} \right]=\int_{-\infty}^\infty {\mathcal{F}_x[b(x)](\omega)\over \mathcal{F}_x[f(x,y)](\omega)}e^{2\pi i \omega x} \mathrm{d}\omega $
it should be Lebesgue measurable:
$ \int_{-\infty}^\infty |b(x)| \, dx < \infty $
$ \int_{-\infty}^\infty |f(x,y)| \, dx < \infty $
One necessary condition is $\mathcal{F}_x[f(x,y)](\omega) \neq 0$. The problem with this method is that Fourier transform is not easy to calculate, though I can use DFT.