On the domain of the function that undergoes the Hilbert transform

[elgen]

Hi all,

This question is on the Hilbert transform, particularly on the domain of the input and output functions of the Hilbert transform.

Before rising the question, consider the Fourier transform. The input is $f(t)$ and the output is $F(\omega)$. The function $f$ and $F$ are defined over different domains, $t$ and $\omega$ respectively.

For the Hilbert transform, the input is $f(t)$ and the output is $\hat{f}(t)$. Both the input and output functions are defined over the same domain. This seems to be inconsistent with the definition of an integral transform.

The question is "would the Hilbert transform be an integral tranform?". My feeling is that the Hilbert transform IS an integral transform. It is a coincidance that the domain of the input function is the same as the domain of the output function. Some assurance is appreciated.

Thank you for the attention.


elgen

 

[Vargo]

Hello.

Yes, it is an integral transform. You are right, the fact that the function and its transform are defined over the same kind of variables (same domain) is coincidental.

See the following references:
http://en.wikipedia.org/wiki/Integral_transform
http://en.wikipedia.org/wiki/Singular_integral

 

[elgen]

Thank you.