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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 chapter mentions that the Fourier transform of a continuous function with compact support can be extended to the complex space C n, as an entire analytic function of exponential type. The chapter also describes the analogy between Paley-Wiener theorem, and the theorem on the Fourier-Borel transformation of the analytic functionals.
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
Generally, the Fourier transform can be defined for any tempered distribution; moreover, any distribution of compact support v is a tempered distribution. If v is a distribution of compact support and f is an infinitely differentiable function, the expression = (↦ ()) is well defined.
https://www.researchgate.net/post/Can_we_say_that_if_Fourier_transform_of_f_has_compact_support_the_Fourier_transform_of_Tf_where_T_is_any_bounded_linear_operator_on_Hilbert_space
Can we say that if Fourier transform of (f) has compact support the Fourier transform of (Tf) where T is any bounded linear operator on Hilbert 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.
https://en.wikipedia.org/wiki/Fourier_transform
There is a close connection between the definition of Fourier series and the Fourier transform for functions f that are zero outside an interval. For such a function, we can calculate its Fourier series on any interval that includes the points where f is not identically zero. The Fourier transform is also defined for such a function.
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