FBI Transform - Inversion Formula

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

f(0)= (2\pi)^{-n/2}\int_{{\mathbf R}^n} {\mathcal F}_a (f)(t,0) \, dt + (2\pi)^{-n/2} \int_{{\mathbf R}^n} \sum_{k=1}^n {2ait_k\over |t|} \cdot {\mathcal F}_a (x_kf)(t,0)\, dt.

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.

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