Editing Hilbert transform (section)

===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>