Fourier inversion theorem
Several different Fourier inversion theorems
exist. One sometimes sees the following identity 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; it may be only "conditionally integrable". For example, the function f
) = 1 if −a
) = 0 otherwise has Fourier transform
In such a case, the integral in the Fourier inversion theorem above must be taken to be an improper integral
rather than a Lebesgue integral.
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
[How that is done might be explained here.
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
The Fourier inversion theorem might then say that
What kind of convergence is right? "Convergence in mean square" can be proved fairly easily:
What about convergence almost everywhere
? That would say that if f
is quadratically integrable, then for "almost every" value of x
between 0 and 2π we have
Perhaps surprisingly, although this result is true, it was not proved until 1966.
For strictly finitary discrete Fourier transforms, these delicate questions of convergence are avoided.