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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?

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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 8 '12 at 9:48
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 8 '12 at 13:38

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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You got me. I usually ask a variant of this question at wedding parties along with "You call that a ring?!" – Tom Copeland Oct 8 '12 at 22:37

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