Fourier Transforms of Square-integrable Functions
Plancherel theorem allows the Fourier transform to be extended to a unitary operator on the Hilbert space of all square-integrable functions, i.e., all functions satisfying
Therefore it is invertible on L2.
In case f is a square-integrable periodic function on the interval, 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 square-integrable, then for "almost every" value of x between 0 and 2π we have
This was not proved until 1966 in (Carleson, 1966).
For strictly finitary discrete Fourier transforms, these delicate questions of convergence are avoided.
Read more about this topic: Fourier Inversion Theorem
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