In mathematics, the FBI transform or Fourier–Bros–Iagolnitzer transform is a generalization of the Fourier transform developed by the French mathematical physicistsJacques Bros and Daniel Iagolnitzer in order to characterise the local analyticity of functions (or distributions) on Rn. The transform provides an alternative approach to analyticwave front sets of distributions, developed independently by the Japanese mathematicians Mikio Sato, Masaki Kashiwara and Takahiro Kawai in their approach to microlocal analysis. It can also be used to prove the analyticity of solutions of analytic elliptic partial differential equations as well as a version of the classical uniqueness theorem, strengthening the Cauchy–Kowalevski theorem, due to the Swedish mathematician Erik Holmgren (1873–1943).
The FBI transform of f is defined for a ≥ 0 by
Thus, when a = 0, it essentially coincides with the Fourier transform.
The same formulas can be used to define the Fourier and FBI transforms of tempered distributions in S'(Rn).
Sumber : http://en.wikipedia.org/wiki/FBI_transform
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