Editing Hilbert transform (section)

==Extending the domain of definition==

===Hilbert transform of distributions===
It is further possible to extend the Hilbert transform to certain spaces of [[distribution (mathematics)|distributions]] {{harv|Pandey|1996|loc=Chapter 3}}. Since the Hilbert transform commutes with differentiation, and is a bounded operator on {{mvar|L<sup>p</sup>}}, {{mvar|H}} restricts to give a continuous transform on the [[inverse limit]] of [[Sobolev spaces]]:

<math display="block">\mathcal{D}_{L^p} = \underset{n \to \infty}{\underset{\longleftarrow}{\lim}} W^{n,p}(\mathbb{R})</math>

The Hilbert transform can then be defined on the dual space of <math>\mathcal{D}_{L^p}</math>, denoted <math>\mathcal{D}_{L^p}'</math>, consisting of {{mvar|L<sup>p</sup>}} distributions. This is accomplished by the duality pairing:<br/>
For {{nowrap|<math> u\in \mathcal{D}'_{L^p} </math>,}} define:

<math display="block">\operatorname{H}(u)\in \mathcal{D}'_{L^p} = \langle \operatorname{H}u, v \rangle \ \triangleq \ \langle u, -\operatorname{H}v\rangle,\ \text{for all} \ v\in\mathcal{D}_{L^p} .</math>

It is possible to define the Hilbert transform on the space of [[tempered distributions]] as well by an approach due to Gel'fand and Shilov,{{sfn|Gel'fand|Shilov|1968}} but considerably more care is needed because of the singularity in the integral.

=== Hilbert transform of bounded functions ===
The Hilbert transform can be defined for functions in <math>L^\infty (\mathbb{R})</math> as well, but it requires some modifications and caveats. Properly understood, the Hilbert transform maps <math>L^\infty (\mathbb{R})</math> to the [[Banach space]] of [[bounded mean oscillation]] (BMO) classes.

Interpreted naïvely, the Hilbert transform of a bounded function is clearly ill-defined. For instance, with {{math|1=''u'' = sgn(''x'')}}, the integral defining {{math|H(''u'')}} diverges almost everywhere to {{math|±∞}}. To alleviate such difficulties, the Hilbert transform of an {{math|''L''<sup>∞</sup>}} function is therefore defined by the following [[regularization (physics)|regularized]] form of the integral

<math display="block">\operatorname{H}(u)(t) = \operatorname{p.v.} \int_{-\infty}^\infty u(\tau)\left\{h(t - \tau)- h_0(-\tau)\right\} \, \mathrm{d}\tau</math>

where as above {{math|1=''h''(''x'') = {{sfrac|1|''πx''}}}} and

<math display="block">h_0(x) = \begin{cases}
0                  & \text{if} ~ |x| < 1 \\
\frac{1}{\pi \, x} & \text{if} ~ |x| \ge 1
\end{cases}</math>

The modified transform {{math|H}} agrees with the original transform up to an additive constant on functions of compact support from a general result by Calderón and Zygmund.<ref>{{harvnb|Calderón|Zygmund|1952}}; see {{harvnb|Fefferman|1971}}.</ref> Furthermore, the resulting integral converges pointwise almost everywhere, and with respect to the BMO norm, to a function of bounded mean oscillation.

A [[deep result]] of Fefferman's work<ref>{{harvnb|Fefferman|1971}}; {{harvnb|Fefferman|Stein|1972}}</ref> is that a function is of bounded mean oscillation if and only if it has the form {{nowrap| {{math|''f'' + H(''g'')}} }} for some {{nowrap|<math> f,g \isin L^\infty (\mathbb{R})</math>.}}