The Plancherel theorem is a fundamental result in the field of harmonic analysis, particularly in the context of Fourier transforms and Fourier series. It establishes an important relationship between the \( L^2 \) spaces of functions and distributions, indicating that the Fourier transform is an isometry on these spaces.
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For the unnormalized angular-frequency transform,After normalization, the Fourier transform extends unitarily to . For integrable and , inversion and Fubini give the identity directly.
Some sources say that this is just the part that says that the norm of a function is the same as the norm of its Fourier transform.
The comment at math.stackexchange.com/questions/446870/bijectiveness-injectiveness-and-surjectiveness-of-fourier-transformation-define/1235725#1235725 may be of interest, it says that the bijection statement is an easy consequence from the norm one, thus the confusion.
TODO does it require it to be in as well? Wikipedia en.wikipedia.org/w/index.php?title=Plancherel_theorem&oldid=987110841 says yes, but courses.maths.ox.ac.uk/node/view_material/53981 does not mention it.