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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?
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.