I have some questions about Hilbert transform when I read Real Analysis:
In Stein "Real Analysis" p.220, the Hilbert transform is defined by $P=\frac{I+iH}{2}$, where $P$ is an orthogonal projection on the subspace $S$ of $L^{2}$. But to my knowledge, the Hilbert transform of a function $f(x)$ is defined to be the convolution of $f(x)$ and $\frac{1}{\pi x}$ in signal processing. I cannot figure out the connection of these two definitions.
In p.221, it mentions "Among the many important properties of Hilbert transform is its connection to conjugate harmonic functions. Indeed, for $f $ a real-valued function in $L^{2}(R)$, $f$ and $H(f)$ are, respectively, the real and imaginary parts of the boundary values of a function in the Hardy space.". Would anyone know where is this result from?
I just know a similar conclusion. In Loukas Grafakos's book Classical Fourier Analysis, he prove that, for function $\varphi$ in $\mathcal S(\Bbb R)$, we have $$ F_{\varphi}(z)=\cfrac i\pi\int_{-\infty}^{\infty}\cfrac{\varphi(t)}{z-t}\mathrm dt \to \varphi(x)+iH(\varphi)(x). $$ And he given a proof that extends $H(f)$ from $\mathcal S$ to $L^p$. When $p\gt1$, we know that $H^p=L^p$. I hope this can help you.