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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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    $\begingroup$ Fourier transforms are a bit like manifolds: hard to define, but you know one when you see one. $\endgroup$
    – Amritanshu Prasad
    Commented Oct 8, 2012 at 9:48
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    $\begingroup$ 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. $\endgroup$
    – user23078
    Commented Oct 8, 2012 at 13:38
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    $\begingroup$ @user23078, a nonzero function and its Fourier transform can't both have compact support. $\endgroup$
    – KConrad
    Commented Oct 29, 2016 at 21:01
  • $\begingroup$ Does the Fourier-Mukai transform become the usual Fourier transform after applying the Abel-Jacobi map ? $\endgroup$
    – reuns
    Commented Dec 31, 2016 at 21:44
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    $\begingroup$ 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. $\endgroup$
    – Tom Copeland
    Commented Jan 11, 2017 at 18:10

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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.

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    $\begingroup$ You got me. I usually ask a variant of this question at wedding parties along with "You call that a ring?!" $\endgroup$
    – Tom Copeland
    Commented Oct 8, 2012 at 22:37
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    $\begingroup$ What if there is no underlying field for linearity, only convolution and product operations? $\endgroup$
    – Michael
    Commented Aug 30, 2016 at 20:28

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