I'm trying to solve the following integral equation to find the function $f(x)$ \begin{equation} f(x) = K(x) - \int_0^\infty K(x-t)f(t)dt \end{equation}

where \begin{equation} K(x) = \sum_{i=1}^N c_i e^{-\kappa_i |x|}, \quad \forall i\quad\kappa_i > 0 \end{equation} with the convenient property $K(x-t)=K(t-x)=K(|x-t|)$.

I know that this is an inhomogeneous Fredholm equation of the second kind, however I'm not sure where to begin solving this. Any references and suggestions where to start would be of great help.

  • $\begingroup$ You need to show that K is a compact operator in the space you are interested in. $\endgroup$ – timur Feb 4 at 15:57
  • $\begingroup$ Ok, I'm not sure what that means but I'm fairly confident that a solution exists. $\endgroup$ – Freelunch Feb 5 at 8:04

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