Fourier Transform and Hilber transform, properties

[kidsasd987]

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?

 

[mathman]

Fourier transform is 1-1. The inverse transform will give you the same function back (except on a set of measure 0).

 

[micromass]

Fourier transform is 1-1.

With what domain and codomain?

 

[mathman]

With what domain and codomain?

Reals and reals.

 

[micromass]

Reals and reals.

So the Fourier transform of a real number is a real number?