# On the domain of the function that undergoes the Hilbert transform

1. Oct 30, 2012

### 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 defintion 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

2. Oct 30, 2012

3. Nov 2, 2012

Thank you.