Fourier Inversion Theorem

Fourier Inversion Theorem

In mathematics, Fourier inversion recovers a function from its Fourier transform. Several different Fourier inversion theorems exist.

The following expression is commonly used as the definition of the Fourier transform:

From this, the following inversion formula is found

In this way, one recovers a function from its Fourier transform.

However, this way of stating a Fourier inversion theorem obscures several potential complications not immediately apparent. One Fourier inversion theorem assumes that is Lebesgue-integrable, i.e., the integral of its absolute value is finite:

In that case, the Fourier transform is not necessarily Lebesgue-integrable. For example, the function if and otherwise has Fourier transform:

In such a case, Fourier inversion theorems will usually deal with the convergence of the integral

By contrast, if is taken to be a tempered distribution -- a type of generalized function -- then its Fourier transform is again a tempered distribution; and the Fourier inversion formula is easier to prove.