# Wave front set from the FBI or Segal-Bergman transform (and a motivation)

In the André Martinez's notes "Introduction to microlocal and semiclassical analysis" the Wave Front Set is defined as the complement of the set of points having neighborhoods where the FBI transform is exponentially small. Similarly, one can substitute "the FBI trasnform is exponentially small" by "an improved bound on the Segal-Bergman holds" transform".

I would like to understand how these definitions are seen to be the same that Hörmander's original definition of the Wave Front Set in terms of the Fourier transform.

Also, some motivation for the definition of the apparently quite unnatural FBI transform would be appreciated.

Thanks.

Jacques Bros and Daniel Iagolnitzer's original motivation to introduce the FBI transform in the 70's was to analyse the analytic wave front set of certain distributions / hyperfunctions that appear in quantum field scattering theory (in their work, they followed Sato's school and called the analytic wave front set of a distribution its essential support, due to the Holmgren-type theorem proven by Kashiwara that links the support and the analytic wave front set of a hyperfunction). When working at the hyperfunction level, where everything is boundary values of holomorphic functions, it's pretty natural to convolute using a Gaussian weight, which is entire analytic and intrinsically linked to the geometry of the complex domain. The holomorphic extension of the Gaussian weight off real arguments $x$ then automatically includes the imaginary exponential appearing in the Fourier transform, as follows:
$$\exp(-\lambda\langle z,z\rangle)=\exp(-\lambda\langle x+i\xi,x+i\xi\rangle)=e^{\lambda\langle \xi,\xi\rangle}e^{-\lambda\langle x,x\rangle}e^{-2i\lambda\langle x,\xi\rangle}\ .$$