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 ${\displaystyle f:\mathbb {R} \to \mathbb {C} }$ 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 ${\displaystyle R}$ is the flip operator i.e. ${\displaystyle (Rf)(x):=f(-x)}$, then

The theorem holds if both ${\displaystyle f}$ and its Fourier transform are absolutely integrable (in the Lebesgue sense) and ${\displaystyle f}$ is continuous at the point ${\displaystyle x}$. 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 ${\displaystyle f}$ is an integrable continuous function. Use the convention for the Fourier transform that