Inversion Formula
The Fourier inversion formula
allows a function f to be recovered from its Fourier transform.
In particular
Similarly. at a positive value of a, f(0) can be recovered from the FBI transform of f(x) and xkf(x) by the inversion formula
This formula can be proved by calculating the coefficients of am in the analytic function of a defined by the right hand side. These coefficients can be expressed in terms of the Fourier transform, the Euler operator
and the Laplacian operator
and easily computed using integration by parts.
Read more about this topic: FBI Transform
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