Fourier transform, more exactly DFT, is a special case of matrix multiplication, when seen as a linear operator. The fact that you can do operations faster than

Fourier transform, more exactly DFT, is a special case of matrix multiplication, when seen as a linear operator. The fact that you can do operations faster than the usual N*2 matrix / vector operation is due to the specific form of the underlying matrix (Van Der Mont matrix). So they are not that different: the specific structure of the Fourier basis allows for more efficient matrix * vector operation.