Two MO questions, "[Heuristic behind the Fourier-Mukai transform][1]" and "[Explaining Mukai-Fourier transforms physically][2]," 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?

  [1]: http://mathoverflow.net/questions/9834/heuristic-behind-the-fourier-mukai-transform
  [2]: http://mathoverflow.net/questions/97361/explaining-mukai-fourier-transforms-physically