Editing Hilbert transform (section)

== Definition ==
The Hilbert transform of {{mvar|u}} can be thought of as the [[convolution]] of {{math|''u''(''t'')}} with the function {{math|1=''h''(''t'') = {{sfrac|1|{{pi}}''t''}}}}, known as the [[Cauchy kernel]]. Because 1/{{mvar|t}} is not [[integrable]] across {{math|1=''t'' = 0}}, the integral defining the convolution does not always converge. Instead, the Hilbert transform is defined using the [[Cauchy principal value]] (denoted here by {{math|p.v.}}). Explicitly, the Hilbert transform of a function (or signal) {{math|''u''(''t'')}} is given by

<math display="block">
 \operatorname{H}(u)(t) = \frac{1}{\pi}\, \operatorname{p.v.} \int_{-\infty}^{+\infty} \frac{u(\tau)}{t - \tau}\,\mathrm{d}\tau,
</math>

provided this integral exists as a principal value. This is precisely the convolution of {{mvar|u}} with the [[Distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution]] {{math|p.v. {{sfrac|1|{{pi}}''t''}}}}.<ref>Due to {{harvnb|Schwartz|1950}}; see {{harvnb|Pandey|1996|loc=Chapter 3}}.</ref> Alternatively, by changing variables, the principal-value integral can be written explicitly<ref>{{harvnb|Zygmund|1968|loc=§XVI.1}}.</ref> as

<math display="block">
 \operatorname{H}(u)(t) = \frac{2}{\pi}\, \lim_{\varepsilon \to 0} \int_\varepsilon^\infty \frac{u(t - \tau) - u(t + \tau)}{2\tau} \,\mathrm{d}\tau.
</math>

When the Hilbert transform is applied twice in succession to a function {{mvar|u}}, the result is

<math display="block">
 \operatorname{H}\bigl(\operatorname{H}(u)\bigr)(t) = -u(t),
</math>

provided the integrals defining both iterations converge in a suitable sense. In particular, the inverse transform is 
<math>-\operatorname{H}</math>. This fact can most easily be seen by considering the effect of the Hilbert transform on the [[Fourier transform]] of {{math|''u''(''t'')}} (see {{slink|#Relationship with the Fourier transform}} below).

For an [[analytic function]] in the [[upper half-plane]], the Hilbert transform describes the relationship between the real part and the imaginary part of the boundary values. That is, if {{math|''f''(''z'')}} is analytic in the upper half complex plane {{math|1={''z'' : Im{''z''} > 0}<nowiki/>}}, and {{math|1=''u''(''t'') = Re{''f'' (''t'' + 0·''i'')} }}, then {{math|1= Im{''f''(''t'' + 0·''i'')} = H(''u'')(''t'')}} up to an additive constant, provided this Hilbert transform exists.

===Notation===
In [[signal processing]] the Hilbert transform of {{math|''u''(''t'')}} is commonly denoted by <math>\hat{u}(t)</math>.<ref>E.g., {{harvnb|Brandwood|2003|p=87}}.</ref> However, in mathematics, this notation is already extensively used to denote the Fourier transform of {{math|''u''(''t'')}}.<ref>E.g., {{harvnb|Stein|Weiss|1971}}.</ref> Occasionally, the Hilbert transform may be denoted by <math>\tilde{u}(t)</math>. Furthermore, many sources define the Hilbert transform as the negative of the one defined here.<ref>E.g., {{harvnb|Bracewell|2000|p=359}}.</ref>