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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. Ask Question ... On functions with Fourier transform having compact support. 1. ... Why must the Fourier transform of a compactly support function not have compact support? 0. Prove that the Fourier transform of a test function has not compact support. 0.
https://www.sciencedirect.com/science/article/pii/S0079816908602779
This chapter discusses the Fourier transforms of distributions with compact support and Paley-Wiener theorem. This chapter considers a continuous function f with compact support in R n.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.
https://mathoverflow.net/questions/29991/fourier-transforms-of-compactly-supported-functions
Then $\hat{f}$ should remain unchanged when convolved with the Fourier transform of $\chi$, but since $\hat{f}$ lives on the reals you would like to think that this convolution would partially fill up some open set connected to the boundary of the original support of $\hat{f}$ and thus enlarge the support of $\hat{f}$.
http://math.uchicago.edu/~may/REU2013/REUPapers/Hill.pdf
and its Fourier transform cannot both be concentrated on small sets. We begin with the basic properties of the Fourier transform and show that a function and its Fourier 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
https://en.wikipedia.org/wiki/Pontryagin_duality
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as , the circle, or finite cyclic groups.The Pontryagin duality theorem itself states that locally compact abelian groups identify naturally with their bidual.
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? ... the Fourier transform of T(f) also has ...
https://en.wikipedia.org/wiki/Paley%E2%80%93Wiener_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 . The formulation presented here is from Hörmander (1976).
https://en.wikipedia.org/wiki/Fourier_transform
The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation.
https://en.wikipedia.org/wiki/Distribution_(mathematics)
Distribution of compact support. It is also possible to define the convolution of two distributions S and T on R n, provided one of them has compact support. Informally, in order to define S∗T where T has compact support, the idea is to extend the definition of the convolution ∗ to a linear operation on distributions so that the ...
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