Several different Fourier inversion theorems exist. One sometimes sees the following identity used as the definition of the 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:
By contrast, if we take f to be a tempered distribution -- a sort of generalized function -- then its Fourier transform is a function of the same sort: another tempered distribution; and the Fourier inversion formula is more simply proved.
One can also define the Fourier transform of a quadratically integrable function, i.e., one satisfying
Then the Fourier transform is another quadratically integrable function.
In case f is a quadratically integrable periodic function on the interval then it has a Fourier series whose coefficients are
For strictly finitary discrete Fourier transforms, these delicate questions of convergence are avoided.