Is it possible to define a "generalized Fourier transform" that corresponds to a given generalized Fourier series - e.g. with respect to Chebyshev or Legendre polynomials, rather than exp(+-inx)? If so, then what would the dual space represent (in the context of say, signals where x represents time), and if not, then what is so special about the basis, exp(+-inx) (or equivalently sin(nx), cos(nx)) that allows such a transform to be defined?
Also, sorry if this is poorly worded - I'm pretty retarded.