Integral equation of the first kind

Summary

An integral equation of the first kind is a linear integral equation in which the unknown function appears only under the integral: $\int K(x, s) \varphi(s) \, ds = f(x)$. Solving it means inverting an integral operator. Such equations are typically ill-posed: a solution may fail to exist, and small changes of $f$ can cause large changes of $\varphi$. Inverse problems, such as recovering a signal from blurred data, lead to equations of the first kind.