Band limited property of Hilbert transform?

[rsingh628]

I’ve read in a signal processing textbook that the Hilbert transform of a band limited function is likewise band limited. How can one prove that property, where the Hilbert transform of a band limited function is also band limited, which I take to mean that the Fourier transform is zero for some positive frequency [imath] |s| > s_0 [/imath] ? Initial thought is that in the frequency domain, the Hilbert transform is the same as multiplying by −isgn(s), but I’m not sure how to carry out the theory. Any help would be appreciated.

 

[blamocur]

Here is an informal sketch of a "proof" :

1. Hilbert transform is a convolution with so called "Cauchy kernel"
2. Fourier transform of a convolution is a product of Fourier transforms of the convolution's arguments.

Details can be found in Wikipedia.