Editing Hilbert transform (section)

===Convolutions===
The Hilbert transform can formally be realized as a [[convolution]] with the [[Distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution]]{{sfn|Duistermaat|Kolk|2010|p=211}}

<math display="block">h(t) = \operatorname{p.v.} \frac{1}{ \pi \, t }</math>

Thus formally,

<math display="block">\operatorname{H}(u) = h*u</math>

However, ''a priori'' this may only be defined for {{mvar|u}} a distribution of [[compact support]]. It is possible to work somewhat rigorously with this since compactly supported functions (which are distributions ''a fortiori'') are [[dense (topology)|dense]] in {{math|''L<sup>p</sup>''}}.  Alternatively, one may use the fact that ''h''(''t'') is the [[distributional derivative]] of the function {{math|1=log{{!}}''t''{{!}}/''π''}}; to wit

<math display="block">\operatorname{H}(u)(t) = \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{\pi} \left(u*\log\bigl|\cdot\bigr|\right)(t)\right)</math>

For most operational purposes the Hilbert transform can be treated as a convolution. For example, in a formal sense, the Hilbert transform of a convolution is the convolution of the Hilbert transform applied on ''only one'' of either of the factors:

<math display="block">\operatorname{H}(u*v) = \operatorname{H}(u)*v = u*\operatorname{H}(v)</math>

This is rigorously true if {{mvar|u}} and {{mvar|v}} are compactly supported distributions since, in that case,

<math display="block"> h*(u*v) = (h*u)*v = u*(h*v)</math>

By passing to an appropriate limit, it is thus also true if {{math|''u'' ∈ ''L<sup>p</sup>''}} and {{math|''v'' ∈ ''L<sup>q</sup>''}} provided that

<math display="block"> 1 < \frac{1}{p} + \frac{1}{q} </math>

from a theorem due to Titchmarsh.{{sfn|Titchmarsh|1948|loc=Theorem 104}}