I'm having trouble solving the following exercise:
Use the Fourier transform to solve the integral equation
$$f(x) = \int_{-\infty}^{\infty} e^{-|x-\xi|}u(\xi)\,d\xi$$
Then verify your solution when $f(x) = e^{-2|x|}.$
My idea was:
Since $F[f(x)] = F[h \ast u(x)] = \sqrt{2\pi} *\hat{h}(k)\hat{u}(k) = \frac{2\hat{u}(k)}{k^2+1}$, then by the Fourier inversion $f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \frac{2\hat{u}(k)}{k^2+1}e^{ikx}dk$.
I'm not sure if this is a correct intuition or not. Does anyone have some hints?