Editing Hilbert transform (section)

===Boundedness===
If {{math|1 < ''p'' < ∞}}, then the Hilbert transform on <math>L^p(\mathbb{R})</math> is a [[bounded linear operator]], meaning that there exists a constant {{mvar|C<sub>p</sub>}} such that

<math display="block">\left\|\operatorname{H}u\right\|_p \le C_p \left\|u\right\|_p </math>

for all {{nowrap|<math>u \isin L^p(\mathbb{R})</math>.}}<ref>This theorem is due to {{harvnb|Riesz|1928|loc=VII}}; see also {{harvnb|Titchmarsh|1948|loc=Theorem 101}}.</ref>

The best constant <math>C_p</math> is given by<ref>This result is due to {{harvnb|Pichorides|1972}}; see also {{harvnb|Grafakos|2004|loc=Remark 4.1.8}}.</ref>
<math display="block">C_p = \begin{cases}
  \tan \frac{\pi}{2p} & \text{if} ~ 1 < p \leq 2 \\[4pt] 
  \cot \frac{\pi}{2p} & \text{if} ~ 2 < p < \infty
\end{cases}</math>

An easy way to find the best <math>C_p</math> for <math>p</math> being a power of 2 is through the so-called Cotlar's identity that <math> (\operatorname{H}f)^2 =f^2 +2\operatorname{H}(f\operatorname{H}f)</math> for all real valued {{mvar|f}}. The same best constants hold for the periodic Hilbert transform.

The boundedness of the Hilbert transform implies the <math>L^p(\mathbb{R})</math> convergence of the symmetric partial sum operator 
<math display="block">S_R f = \int_{-R}^R \hat{f}(\xi) e^{2\pi i x\xi} \, \mathrm{d}\xi </math>

to {{mvar|f}} in {{nowrap|<math>L^p(\mathbb{R})</math>.}}<ref>See for example {{harvnb|Duoandikoetxea|2000|p=59}}.</ref>