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https://math.stackexchange.com/questions/526449/fourier-transform-of-a-function-of-compact-support
Fourier transform of a function of compact support. My professor occasionally assigns optional difficult problems which we do not turn in from Stein and Shakarchi's Complex Analysis. I am currently studying for a test in that class and try to get all of these optional problems answered.
https://www.sciencedirect.com/science/article/pii/S0079816908602779
The Fourier transform of a distribution T with compact support in Rn is the function, in Rn , (29.4) T{£) = (TX, exp(-2*V <*,£>)>• T can be extended to the complex space Cn as an entire analytic, given by (29.5) f({) = <r,, exp(-2wr <*,£»>• Proof.
https://mathoverflow.net/questions/29991/fourier-transforms-of-compactly-supported-functions
For R. Suppose f is our compactly supported function and g(x) is its Fourier transform. Since f is compactly supported, ˆf = g is the restriction to R of an entire function g(z) by the Paley-Wiener theorems. Since g is entire and vanishes on an open set, g ≡ 0. The proof of this last fact...
http://math.uchicago.edu/~may/REU2013/REUPapers/Hill.pdf
transform cannot both have compact support. From there we prove the Fourier inversion theorem and use this to prove the classical uncertainty principle which shows that the spread of a function and its Fourier transform are inversely proportional. Finally, we extend our compactness result from earlier and show that a function and its Fourier transform cannot both be supported on nite sets. Contents …
https://en.wikipedia.org/wiki/Paley-Wiener_theorem
Schwartz's Paley–Wiener theorem asserts that the Fourier transform of a distribution of compact support on R n is an entire function on C n and gives estimates on its growth at infinity. It was proven by Laurent Schwartz (1952). The formulation presented here is from Hörmander (1976).
https://www.encyclopediaofmath.org/index.php/Fourier_transform
The inversion formula for the Fourier transform is very simple: Under the action of the Fourier transform linear operators on the original space, which are invariant with respect to a shift, become (under certain conditions) multiplication operators in the image space.
https://en.wikipedia.org/wiki/Talk:Convergence_of_Fourier_series
Musical signals, for example, can be seen as functions with finite energy (L^1 and L^2)(is a function of the time which has a compact support and is in L^00) and a compact spectrum so it’s possible to represent their information (which has the power of the continuum) with discrete series.
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