Fredholm equation of the first kind

Summary

A Fredholm equation of the first kind is an integral equation $\int_a^b K(x, y) \varphi(y) \, dy = f(x)$ with fixed limits $a$ and $b$. The unknown function $\varphi$ appears only under the integral sign. It is the Fredholm integral equation form of an integral equation of the first kind. Such equations model the recovery of a cause from a smoothed measurement, as in deblurring and tomography. They are usually ill-posed: a solution may fail to exist, and small changes of $f$ can cause large changes of $\varphi$.