Fourier transform of tempered distributions

Summary

The Fourier transform of a tempered distribution $T$ is again a tempered distribution. It acts on a Schwartz function $\varphi$ as $T$ acts on the Fourier transform of $\varphi$. It extends the Fourier transform from integrable and square-integrable functions to all tempered distributions. It is a linear bijection of the space of tempered distributions onto itself. It turns derivatives into multiplication by polynomials, which makes it a main tool for linear partial differential equations.