Integral transform

Summary

An integral transform turns a function $f$ into a new function $Tf$ by integrating $f$ against a fixed function of two variables, the kernel. In symbols, $(Tf)(u) = \int f(t) K(t, u)\, dt$. The Laplace transform, the Fourier transform and the Mellin transform are the main examples. A good transform turns a hard operation, such as differentiation or convolution, into an easy one, such as multiplication. It can be undone by an inverse transform.