Tempered distribution

Summary

A tempered distribution on $\R^n$ is a continuous linear functional on the Schwartz space of smooth, rapidly decreasing functions. Tempered distributions include all $L^p$ functions, all polynomials, finite measures and the Dirac delta. They form the natural domain of the Fourier transform, which maps the space of tempered distributions onto itself. For this reason they are the standard setting for linear partial differential equations with constant coefficients and for function spaces defined by Fourier analysis.