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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 Asked 6 years, 2 months ago. ... On functions with Fourier transform having compact support. 1. ... Prove that the Fourier transform of a test function has not compact support. 0.
https://math.stackexchange.com/questions/154454/a-function-and-its-fourier-transform-cannot-both-be-compactly-supported
A funtion and its fourier transformation cannot both be compactly supported unless f=0 1 If a function is compactly supported, then its Fourier series converge?
https://mathoverflow.net/questions/29991/fourier-transforms-of-compactly-supported-functions
For $\mathbb{R}$. Suppose f is our compactly supported function and g(x) is its Fourier transform. Since f is compactly supported, $\hat{f} = g$ is the restriction to $\mathbb{R}$ of an entire function g(z) by the Paley-Wiener theorems.
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://en.wikipedia.org/wiki/Fourier_transform
The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an uncertainty principle by viewing a function and its Fourier transform as conjugate variables with respect to the symplectic form on the time–frequency domain: from the point of view of the linear canonical transformation, the ...
https://en.wikipedia.org/wiki/Paley%E2%80%93Wiener_theorem
and that this function can be extended to values of s in the complex space C n. This extension of the Fourier transform to the complex domain is called the Fourier–Laplace transform. Schwartz's theorem. An entire function F on C n is the Fourier–Laplace transform …
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://ui.adsabs.harvard.edu/abs/1999PhDT........75N/abstract
Two theorems establish the bounds on the norm of a real even continuous perturbations of the top hat function such that the zeros of the Fourier transform of the perturbed function remain real and complete. A necessary condition such that a real function of compact support has a Fourier transform with all real zeros has been established.Author: Arjang Jaden Noushin
https://www.numericana.com/answer/fourier.htm
So, Schwartz introduced a larger space of test functions, stable under Fourier transform, whose duals are called "tempered distributions" for which the Fourier transform is well-defined by duality, as explained below. The support of a function is the closure of the set of all points for which it's nonzero.
https://www.osti.gov/pages/biblio/1427516-fast-algorithm-convolution-functions-compact-support-using-fourier-extensions
Abstract. In this paper, we present a new algorithm for computing the convolution of two compactly supported functions. The algorithm approximates the functions to be convolved using Fourier extensions and then uses the fast Fourier transform to efficiently compute Fourier extension approximations to the pieces of the result.
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