Textbook says, Fourier transform expresses a function in time domain as a function in frequency domain. Basically, fourier transform gives two different expressions in terms of t domain and f domain but they represent the same signal. It also says Hilbert transform is a different type of transform because it gives a new signal not equal to the previous one. However, I do not think Fourier transform does 1 to 1 mapping since we have to integrate t for -infinity to +infinity. (ex, t=1 converts to f=constant.) I understand F.T shifts the domain, but strictly speaking, I don't think they have exactly the same structure (function-wise). In contrast, Hilber transform does 1 to 1 mapping as it shifts the original function by 90 degree. In this case function structure is conserved. I wonder why we say Hilbert transform is a totally different signal from the original one. is it because they are physically different?