Editing Hilbert transform (section)

== Hilbert transform on the circle ==
{{see also|Hardy space}}
For a periodic function {{mvar|f}} the circular Hilbert transform is defined:

<math display="block">\tilde f(x) \triangleq \frac{1}{ 2\pi } \operatorname{p.v.} \int_0^{2\pi} f(t)\,\cot\left(\frac{ x - t }{2}\right)\,\mathrm{d}t</math>

The circular Hilbert transform is used in giving a characterization of Hardy space and in the study of the conjugate function in Fourier series. The kernel, 
<math display="block">\cot\left(\frac{ x - t }{2}\right)</math>
is known as the '''Hilbert kernel''' since it was in this form the Hilbert transform was originally studied.{{sfn|Khvedelidze|2001}}

The Hilbert kernel (for the circular Hilbert transform) can be obtained by making the Cauchy kernel {{frac|1|{{mvar|x}}}} periodic. More precisely, for {{math|''x'' ≠ 0}}

<math display="block">\frac{1}{\,2\,}\cot\left(\frac{x}{2}\right) = \frac{1}{x} + \sum_{n=1}^\infty \left(\frac{1}{x + 2n\pi} + \frac{1}{\,x - 2n\pi\,} \right)</math>

Many results about the circular Hilbert transform may be derived from the corresponding results for the Hilbert transform from this correspondence.

Another more direct connection is provided by the Cayley transform {{math|1=''C''(''x'') = (''x'' – ''i'') / (''x'' + ''i'')}}, which carries the real line onto the circle and the upper half plane onto the unit disk. It induces a unitary map

<math display="block"> U\,f(x) = \frac{1}{(x + i)\,\sqrt{\pi}} \, f\left(C\left(x\right)\right) </math>

of {{math|''L''<sup>2</sup>('''T''')}} onto <math>L^2 (\mathbb{R}).</math> The operator {{mvar|U}} carries the Hardy space {{math|''H''<sup>2</sup>('''T''')}} onto the Hardy space <math>H^2(\mathbb{R})</math>.{{sfn|Rosenblum|Rovnyak|1997|p=92}}