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(
x) = 1 if −
a <
x <
a and
f(
x) = 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.