Editing Hilbert transform (section)

== Properties ==

===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>

===Anti-self adjointness===
The Hilbert transform is an anti-[[self adjoint]] operator relative to the duality pairing between <math>L^p(\mathbb{R})</math> and the dual space {{nowrap|<math>L^q(\mathbb{R})</math>,}} where {{mvar|p}} and {{mvar|q}} are [[Hölder conjugate]]s and {{math|1 < ''p'', ''q'' < ∞}}. Symbolically,

<math display="block">\langle \operatorname{H} u, v \rangle = \langle u, -\operatorname{H} v \rangle</math>

for <math>u \isin L^p(\mathbb{R})</math> and {{nowrap|<math>v \isin L^q(\mathbb{R})</math>.}}{{sfn|Titchmarsh|1948|loc=Theorem 102}}

===Inverse transform===
The Hilbert transform is an [[anti-involution]],{{sfn|Titchmarsh|1948|p=120}} meaning that

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

provided each transform is well-defined. Since {{math|H}} preserves the space {{nowrap|<math>L^p(\mathbb{R})</math>,}} this implies in particular that the Hilbert transform is invertible on {{nowrap|<math>L^p(\mathbb{R})</math>,}} and that

<math display="block">\operatorname{H}^{-1} = -\operatorname{H}</math>

===Complex structure===
Because {{math|1=H<sup>2</sup> = −I}}  ("{{math|I}}" is the [[identity operator]]) on the real [[Banach space]] of ''real''-valued functions in {{nowrap|<math>L^p(\mathbb{R})</math>,}} the Hilbert transform defines a [[linear complex structure]] on this Banach space. In particular, when {{math|1=''p'' = 2}}, the Hilbert transform gives the Hilbert space of real-valued functions in <math>L^2(\mathbb{R})</math> the structure of a ''complex'' Hilbert space.

The (complex) [[eigenstate]]s of the Hilbert transform admit representations as [[holomorphic function]]s in the upper and lower half-planes in the [[Hardy space]] [[H square|{{math|H<sup>2</sup>}}]] by the [[Paley–Wiener theorem]].

===Differentiation===
Formally, the derivative of the Hilbert transform is the Hilbert transform of the derivative, i.e. these two linear operators commute:

<math display="block">\operatorname{H}\left(\frac{ \mathrm{d}u}{\mathrm{d}t}\right) = \frac{\mathrm d}{\mathrm{d}t}\operatorname{H}(u)</math>

Iterating this identity,

<math display="block">\operatorname{H}\left(\frac{\mathrm{d}^ku}{\mathrm{d}t^k}\right) = \frac{\mathrm{d}^k}{\mathrm{d}t^k}\operatorname{H}(u)</math>

This is rigorously true as stated provided {{mvar|u}} and its first {{mvar|k}} derivatives belong to {{nowrap|<math>L^p(\mathbb{R})</math>.}}{{sfn|Pandey|1996|loc=§3.3}} One can check this easily in the frequency domain, where differentiation becomes multiplication by {{mvar|ω}}.

===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}}

===Invariance===
The Hilbert transform has the following invariance properties on <math>L^2(\mathbb{R})</math>.
* It commutes with translations.  That is, it commutes with the operators {{math|1=''T''<sub>''a''</sub> ''f''(''x'') = ''f''(''x'' + ''a'')}} for all {{mvar|a}} in <math>\mathbb{R}.</math>
* It commutes with positive dilations.  That is it commutes with the operators {{math|1=''M<sub>λ</sub> f'' (''x'') = ''f'' (''λ x'')}} for all {{math|''λ'' > 0}}.
* It [[Anticommutativity|anticommutes]] with the reflection {{math|1=''R f'' (''x'') = ''f'' (−''x'')}}.

Up to a multiplicative constant, the Hilbert transform is the only bounded operator on {{mvar|L}}<sup>2</sup> with these properties.{{sfn|Stein|1970|loc=§III.1}}

In fact there is a wider set of operators that commute with the Hilbert transform. The group <math>\text{SL}(2,\mathbb{R})</math> acts by unitary operators {{math|U<sub>''g''</sub>}} on the space <math>L^2(\mathbb{R})</math> by the formula

<math display="block">\operatorname{U}_{g}^{-1} f(x) = \frac{1}{ c x + d } \, f \left( \frac{ ax + b }{ cx + d } \right) \,,\qquad g = \begin{bmatrix} a & b \\ c & d \end{bmatrix} ~,\qquad \text{ for }~ a d - b c = \pm 1 . </math>
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This [[unitary representation]] is an example of a [[principal series representation]] of <math>~\text{SL}(2,\mathbb{R})~.</math> In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces, [[Hardy space]] <math>H^2(\mathbb{R})</math> and its conjugate. These are the spaces of {{math|''L''<sup>2</sup>}} boundary values of holomorphic functions on the upper and lower halfplanes. <math>H^2(\mathbb{R})</math> and its conjugate consist of exactly those {{math|''L''<sup>2</sup>}} functions with Fourier transforms vanishing on the negative and positive parts of the real axis respectively. Since the Hilbert transform is equal to {{math|1=H = −''i'' (2''P'' − I)}}, with {{mvar|P}} being the orthogonal projection from <math>L^2(\mathbb{R})</math> onto <math>\operatorname{H}^2(\mathbb{R}),</math> and {{math|I}} the [[identity operator]], it follows that <math>\operatorname{H}^2(\mathbb{R})</math> and its orthogonal complement are eigenspaces of {{math|H}} for the eigenvalues {{math|±''i''}}. In other words, {{math|H}} commutes with the operators {{mvar|U<sub>g</sub>}}. The restrictions of the operators {{mvar|U<sub>g</sub>}} to <math>\operatorname{H}^2(\mathbb{R})</math> and its conjugate give irreducible representations of <math>\text{SL}(2,\mathbb{R})</math> – the so-called [[limit of discrete series representation]]s.<ref>See {{harvnb|Bargmann|1947}}, {{harvnb|Lang|1985}}, and {{harvnb|Sugiura|1990}}.</ref>