Integral transforms of tempered distributions

Summary

Many integral transforms extend from functions to tempered distributions by duality. One moves the transform onto the Schwartz test functions. The transform $Tu$ of a distribution $u$ acts on a test function $\varphi$ as $u$ acts on $T^t\varphi$. Here $T^t$ is the transform with the transposed kernel. This works when $T^t$ maps the Schwartz space continuously into itself. The model case is the Fourier transform of tempered distributions; the Laplace transform of distributions uses a related construction.