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Let $f:\mathcal{D}\to\mathcal{D}$ be a function whose domain and co-domain are $\mathcal{D}$. Let $\hat{f}$ be its Hilbert transform, which is defined as

$\hat{f}(t)=\mathcal{H}(f(t))=\frac{1}{\pi} \mathop{p.v.}\int_{-\infty}^{\infty} \frac{f(\tau)}{t-\tau}\ d\tau.$

Now I can see that the domain of $\hat{f}(t)$ is $\mathcal{D}$. What is its co-domain? Is it $\mathcal{D}$ too or can it be different?

NOTE: When I say $\mathcal{D}$, I mean either $\mathbb{R}$ or $\mathbb{C}$

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Actually there is no reason to think that integral converges (even in the principal value sense) for all $t \in \cal D$, so the domain of $\hat{f}$ is not necessarily $\cal D$. On the other hand, if ${\cal D} = \mathbb R$ I see no reason not to allow $t \in \mathbb C$. What is true is that if $f$ is real-valued and $t \in \mathbb R$ and the principal value integral converges, then it will be real.

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    I still don't get the last line.. why should the p.v. converge to a real value?2011-06-18