FBI transform

Summary

The FBI transform, named after Fourier, Bros and Iagolnitzer, is a phase-space transform. It looks at a function near a point $x$ at frequency $\xi$ through a Gaussian window of width about $\abs{\xi}^{-1/2}$. It is a tool of microlocal analysis. Smoothness and real analyticity of a function near a point, in given directions, can be read off from the decay of its FBI transform. In particular, it characterizes the analytic wave front set through exponential decay. It is a close relative of the Bargmann transform and of the short-time Fourier transform with a Gaussian window.