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https://math.stackexchange.com/questions/1571532/fourier-transforms-having-compact-support
Since the Fourier transform maps the Schwarz space to itself there exists a Schwarz function $\phi$ with $\int\phi=1$ such that $\hat\phi$ has compact support. (You can make $\hat\phi$ be any infinitely differentiable function with compact support.)
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 $\hat{f}$.
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://www.physicsforums.com/threads/sufficient-condition-for-bounded-fourier-transform.773050/
Oct 01, 2014 · - Let's suppose F is not bounded and has compact support - Then there must be at least one point in the support of F where the Fourier transform goes to infinity. - Let u be that point, and translate the Fourier transform so that u coincides with the origin. - This means that the inverse FT of the translated F must have integral=infinity.
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/Talk:Convergence_of_Fourier_series
The Fourier transform for jpegs and mp3s can be viewed strictly in the discrete context, in which case the convergence is moot (since the Fourier series is a finite sum.) The remaining comments do not pertain to the convergence of Fourier series. Loisel 14:06, 24 Mar 2005 (UTC)
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).
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