Fourier inversion
In mathematics, Fourier inversion recovers a function from its Fourier transform. Several different Fourier inversion theorems exist.
Sometimes the following expression is used as the definition of the Fourier transform:
Then it is asserted that
In this way, one recovers a function from its Fourier transform.
However, this way of stating a Fourier inversion theorem sweeps some more subtle issues under the carpet. One Fourier inversion theorem assumes that f 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 f(x) = 1 if −a < x < a and f(x) = 0 otherwise has Fourier transform
In such a case, Fourier inversion theorems usually investigate the convergence of the integral
By contrast, if we take f to be a tempered distribution -- a type of generalized function -- then its Fourier transform is another tempered distribution; and the Fourier inversion formula is then more simple to prove.
