Fourier inversion theorem
Mathematical theorem about functions / From Wikipedia, the free encyclopedia
Dear Wikiwand AI, let's keep it short by simply answering these key questions:
Can you list the top facts and stats about Inverse Fourier transform?
Summarize this article for a 10 year old
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.
This article needs additional citations for verification. (May 2024) |
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.