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This question has been revised. Skip to the question in bold.

Two MO questions, "Heuristic behind the Fourier-Mukai transform" and "Explaining Mukai-Fourier transforms physically," compel me to ask these two related questions:

1) What properties do you feel are essential for a transform to possess to be called a "Fourier" transform?

2) What properties of the classical Fourier transform are not necessarily shared by a generalized "Fourier" transform?

In other words, how can I recognize a "Fourier" transform?

Revised question:

What properties do the classical Fourier transform and the Fourier-Mukai share and which do they not?

In Lin's answer to the MO-Q on Heuristics noted above, he states, "I really know almost nothing about the classical Fourier transform, but one of the main points is that the Fourier transform is supposed to be an invertible operation." Perhaps someone familiar with both transforms can fill in that rather large gap in knowledge.

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Tom Copeland
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    Fourier transforms are a bit like manifolds: hard to define, but you know one when you see one. – Amritanshu Prasad Oct 08 '12 at 09:48
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    An important property of the fourier transform is the so-called uncertainty principle,$|xf(x)|{L^2}|\xi \hat{f}(\xi)|{L^2}\ge \frac{1}{4\pi}$ for $|f|_{2}=1$.Another similar property is that a function and its fourier can't both have compact surpport. – user23078 Oct 08 '12 at 13:38
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    @user23078, a nonzero function and its Fourier transform can't both have compact support. – KConrad Oct 29 '16 at 21:01
  • Does the Fourier-Mukai transform become the usual Fourier transform after applying the Abel-Jacobi map ? – reuns Dec 31 '16 at 21:44
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    Three reopen votes have been deleted. As it stands now, this seems a well-focused question, yet I doubt if many users here have the knowledge in the two domains to address the question. In any event, I think I'll start calling any transforms I come up with the Fourier--Hardy--Mellin-Heaviside-Copeland transform since they usually can be connected to invertible convolutions which can be related to all these names. Maybe I should throw in Lie for good measure since the convolutions have group properties. – Tom Copeland Jan 11 '17 at 18:10
  • A link to a current, popular question pretty much fitting the alleged reason for closing this question has been deleted by the censors here. – Tom Copeland Jan 31 '17 at 19:53
  • Related https://golem.ph.utexas.edu/category/2010/11/integral_transforms_and_pullpu.html – Tom Copeland May 02 '21 at 19:38
  • A response in 2019 to the "Heuristic ..." MO-Q (link in the original post) goes a good way to answering this question. – Tom Copeland Oct 17 '21 at 02:10

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This is not a mathematical question, really. On my opinion the main properties is linearity and transforming a convolution into a product.

Alexandre Eremenko
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