Expository article on the Hilbert transform?

Question

Is there a good expository article on the Hilbert transform, explaining the basic properties and the applications that serve as motivation for considering the transform worthy of attention?

It's named after Hilbert, so it is worthy of attention by definition ;-) — tired, Oct 10 '15 at 17:04
@tired : But some idiot whom Hilbert never heard of could introduce a concept and decide to name it after Hilbert in order to get attention for it. ${}\qquad{}$ — Michael Hardy, Oct 10 '15 at 17:05
I heard the following asserted yesterday. Let $u$ be a function on $\mathbb R$. Extend it harmonically to the upper half-plane. Find the harmonic conjugate $v$. The restriction of $v$ to $\mathbb R$ is the Hilbert transform of $u$. ${}\qquad{}$ — Michael Hardy, Oct 10 '15 at 17:07
there is actually a whole mathematical physics book with the authors Hilbert and Courant, where Hilbert wrote indeed not a single back. So your caution is well justified. — tired, Oct 10 '15 at 17:08
Btw: One thing why Hilbert transforms are interesting are because in physics they are very closely related to causality conditions in a given (electromagnetic) problem. We can use them to calculate real part of response functions from imaginary parts and vice versa. Look for "Kramers Kronig relations" if you want to know more. — tired, Oct 10 '15 at 17:11
Yes, under suitable growth conditions on $u$, that's sort of the point. — David C. Ullrich, Oct 10 '15 at 17:23