Two MO questions, "Heuristic behind the Fourier-Mukai transform" and "Explaining Mukai-Fourier transforms physically," compel me to ask these two related questions:
What properties do you feel are essential for a transform to possess to be called a "Fourier" transform?
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?