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Concept
Fourier inversion theorem
Summary
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.
Another way to state the theorem is that if is the flip operator i.e. , then
The theorem holds if both and its Fourier transform are absolutely integrable (in the Lebesgue sense) and is continuous at the point . However, even under more general conditions versions of the Fourier inversion theorem hold. In these cases the integrals above may not converge in an ordinary sense.
In this section we assume that is an integrable continuous function. Use the convention for the Fourier transform that
Furthermore, we assume that the Fourier transform is also integrable.
The most common statement of the Fourier inversion theorem is to state the inverse transform as an integral. For any integrable function and all set
Then for all we have
The theorem can be restated as
If f is real valued then by taking the real part of each side of the above we obtain
For any function define the flip operator by
Then we may instead define
It is immediate from the definition of the Fourier transform and the flip operator that both and match the integral definition of , and in particular are equal to each other and satisfy .
Since we have and
The form of the Fourier inversion theorem stated above, as is common, is that
In other words, is a left inverse for the Fourier transform. However it is also a right inverse for the Fourier transform i.e.
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