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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.