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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://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://math.stackexchange.com/questions/2698819/fourier-transform-with-compact-support
$\begingroup$ Yes, of course, but only the most elementary one, the Dirac distribution which is usually met rather soon in lectures on Fourier Transform, at least for students that are in engineering studies. Your answer [+1] has the advantage to use only Fourier …
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
http://lpsa.swarthmore.edu/Fourier/Series/DerFS.html
The derivation of the Fourier series coefficients is not complete because, as part of our proof, ... A more compact representation of the Fourier Series uses complex exponentials. In this case we end up with the following synthesis and analysis equations: ... notation is similar to that of the Fourier Transform ...
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)(Rated B-class, Mid-importance): WikiProject Mathematics
https://en.wikipedia.org/wiki/Fourrier_transform
for any real number x. The statement that f can be reconstructed from ^ is known as the Fourier inversion theorem, and was first introduced in Fourier's Analytical Theory of Heat, although what would be considered a proof by modern standards was not given until much later. The functions f and ^ often are referred to as a Fourier integral pair or Fourier transform pair.
https://ui.adsabs.harvard.edu/abs/1999PhDT........75N/abstract
This work is an attempt to infer information about the nature of the zeros of the Fourier transform of continuous functions of compact support from the nature of coefficients of their polynomial approximation. Although the main thrust of this work has been directed towards the real functions of compact support, some simple complex cases are also considered. The concept of completeness of a ...Author: Arjang Jaden Noushin
https://www.numericana.com/answer/fourier.htm
Such distributions turn out to be too general in the context of Fourier analysis because the Fourier transform of a function of compact support is never itself a function of compact support. So ... = e-p x 2 is its own Fourier transform. Proof : ... The spectrum of a distribution is the support of its Fourier transform.
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