Fourier inversion theorem

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.