The transform provides an alternative approach to analytic wave front sets of distributions, developed independently by the Japanese mathematicians Mikio Sato, Masaki Kashiwara and Takahiro Kawai in their approach to microlocal analysis.
In particular Similarly, at a positive value of a, f(0) can be recovered from the FBI transform of f(x) by the inversion formula Bros and Iagolnitzer showed that a distribution f is locally equal to a real analytic function at y, in the direction ξ if and only if its FBI transform satisfies an inequality of the form for |ξ| sufficiently large.
A simple consequence of the Bros and Iagolnitzer characterisation of local analyticity is the following regularity result of Lars Hörmander and Mikio Sato (Sjöstrand (1982)).
Let P be an elliptic partial differential operator with analytic coefficients defined on an open subset X of Rn.
The analytic wave front set or singular spectrum WFA(f) of a distribution f (or more generally of a hyperfunction) can be defined in terms of the FBI transform (Hörmander (1983)) as the complement of the conical set of points (x, λ ξ) (λ > 0) such that the FBI transform satisfies the Bros–Iagolnitzer inequality for y the point at which one would like to test for analyticity, and |ξ| sufficiently large and pointing in the direction one would like to look for the wave front, that is, the direction at which the singularity at y, if it exists, propagates.