Fourier Transform Intro - Fourier inversion theorem

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Fourier inversion theorem

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In
mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its

Fourier transform. Intuitively it may be viewed as the statement that if we know all

frequency and
phase information about a wave then we may reconstruct the original wave precisely.

The theorem says that if we have a function

satisfying certain conditions, and we
use the
convention for the Fourier transform that



then



In other words, the theorem says that



This last equation is called the Fourier integral theorem.