Editing Hilbert transform (section)

==Conjugate functions==
The Hilbert transform can be understood in terms of a pair of functions {{math|''f''(''x'')}} and {{math|''g''(''x'')}} such that the function
<math display="block">F(x) = f(x) + i\,g(x)</math>
is the boundary value of a [[holomorphic function]] {{math|''F''(''z'')}} in the upper half-plane.{{sfn|Titchmarsh|1948|loc=Chapter V}} Under these circumstances, if {{mvar|f}} and {{mvar|g}} are sufficiently integrable, then one is the Hilbert transform of the other.

Suppose that <math>f \isin L^p(\mathbb{R}).</math> Then, by the theory of the [[Poisson integral]], {{mvar|f}} admits a unique harmonic extension into the upper half-plane, and this extension is given by

<math display="block">u(x + iy) = u(x, y) = \frac{1}{\pi} \int_{-\infty}^\infty f(s)\;\frac{y}{(x - s)^2 + y^2} \; \mathrm{d}s</math>

which is the convolution of {{mvar|f}} with the [[Poisson kernel]]

<math display="block">P(x, y) = \frac{ y }{ \pi\, \left( x^2 + y^2 \right) }</math>

Furthermore, there is a unique harmonic function {{mvar|v}} defined in the upper half-plane such that {{math|1=''F''(''z'') = ''u''(''z'') + ''i v''(''z'')}} is holomorphic and
<math display="block">\lim_{y \to \infty} v\,(x + i\,y) = 0</math>

This harmonic function is obtained from {{mvar|f}} by taking a convolution with the ''conjugate Poisson kernel''

<math display="block">Q(x, y) = \frac{ x }{ \pi\, \left(x^2 + y^2\right) } .</math>

Thus
<math display="block">v(x, y) = \frac{1}{\pi}\int_{-\infty}^\infty f(s)\;\frac{x - s}{\,(x - s)^2 + y^2\,}\;\mathrm{d}s .</math>

Indeed, the real and imaginary parts of the Cauchy kernel are
<math display="block">\frac{i}{\pi\,z} = P(x, y) + i\,Q(x, y)</math>

so that {{math|1=''F'' = ''u'' + ''i v''}} is holomorphic by [[Cauchy's integral formula]].

The function {{mvar|v}} obtained from {{mvar|u}} in this way is called the [[harmonic conjugate]] of {{mvar|u}}. The (non-tangential) boundary limit of {{math|''v''(''x'',''y'')}} as {{math|''y'' → 0}} is the Hilbert transform of {{mvar|f}}. Thus, succinctly,
<math display="block">\operatorname{H}(f) = \lim_{y \to 0} Q(-, y) \star f</math>

=== Titchmarsh's theorem ===

Titchmarsh's theorem (named for [[Edward Charles Titchmarsh|E. C. Titchmarsh]] who included it in his 1937 work) makes precise the relationship between the boundary values of holomorphic functions in the upper half-plane and the Hilbert transform.{{sfn|Titchmarsh|1948|loc=Theorem 95}} It gives necessary and sufficient conditions for a complex-valued [[square-integrable]] function {{math|''F''(''x'')}} on the real line to be the boundary value of a function in the [[Hardy space]] {{math|H<sup>2</sup>(''U'')}} of holomorphic functions in the upper half-plane {{mvar|U}}.

The theorem states that the following conditions for a complex-valued square-integrable function <math>F : \mathbb{R} \to \mathbb{C}</math> are equivalent:

* {{math|''F''(''x'')}} is the limit as {{math|''z'' → ''x''}} of a holomorphic function {{math|''F''(''z'')}} in the upper half-plane such that <math display="block"> \int_{-\infty}^\infty |F(x + i\,y)|^2\;\mathrm{d}x < K </math>
* The real and imaginary parts of {{math|''F''(''x'')}} are Hilbert transforms of each other.
* The [[Fourier transform]] <math>\mathcal{F}(F)(x)</math> vanishes for {{math|''x'' < 0}}.

A weaker result is true for functions of class {{mvar|[[Lp space|L<sup>p</sup>]]}} for {{math|''p'' > 1}}.{{sfn|Titchmarsh|1948|loc=Theorem 103}} Specifically, if {{math|''F''(''z'')}} is a holomorphic function such that

<math display="block">\int_{-\infty}^\infty |F(x + i\,y)|^p\;\mathrm{d}x < K </math>

for all {{mvar|y}}, then there is a complex-valued function {{math|''F''(''x'')}} in <math>L^p(\mathbb{R})</math> such that {{math|''F''(''x'' + ''i y'') → ''F''(''x'')}} in the {{mvar|L<sup>p</sup>}} norm as {{math|''y'' → 0}} (as well as holding pointwise [[almost everywhere]]).  Furthermore,

<math display="block">F(x) = f(x) + i\,g(x)</math>

where {{mvar|f}} is a real-valued function in <math>L^p(\mathbb{R})</math> and {{mvar|g}} is the Hilbert transform (of class {{mvar|L<sup>p</sup>}}) of {{mvar|f}}.

This is not true in the case {{math|1=''p'' = 1}}. In fact, the Hilbert transform of an {{math|''L''<sup>1</sup>}} function {{mvar|f}} need not converge in the mean to another {{math|''L''<sup>1</sup>}} function. Nevertheless,{{sfn|Titchmarsh|1948|loc=Theorem 105}} the Hilbert transform of {{mvar|f}} does converge almost everywhere to a finite function {{mvar|g}} such that

<math display="block">\int_{-\infty}^\infty \frac{ |g(x)|^p }{ 1 + x^2 } \; \mathrm{d}x < \infty</math>

This result is directly analogous to one by [[Andrey Kolmogorov]] for Hardy functions in the disc.{{sfn|Duren|1970|loc=Theorem 4.2}} Although usually called Titchmarsh's theorem, the result aggregates much work of others, including Hardy, Paley and Wiener (see [[Paley–Wiener theorem]]), as well as work by Riesz, Hille, and Tamarkin<ref>see {{harvnb|King|2009a|loc=§&nbsp;4.22}}.</ref>

=== Riemann–Hilbert problem ===
One form of the [[Riemann–Hilbert problem]] seeks to identify pairs of functions {{math|''F''<sub>+</sub>}} and {{math|''F''<sub>−</sub>}} such that {{math|''F''<sub>+</sub>}} is [[holomorphic function|holomorphic]] on the upper half-plane and {{math|''F''<sub>−</sub>}} is holomorphic on the lower half-plane, such that for {{mvar|x}} along the real axis,
<math display="block">F_{+}(x) - F_{-}(x) = f(x)</math>

where {{math|''f''(''x'')}} is some given real-valued function of {{nowrap|<math>x \isin \mathbb{R}</math>.}} The left-hand side of this equation may be understood either as the difference of the limits of {{math|''F''<sub>±</sub>}} from the appropriate half-planes, or as a [[hyperfunction]] distribution. Two functions of this form are a solution of the Riemann–Hilbert problem.

Formally, if {{math|''F''<sub>±</sub>}} solve the Riemann–Hilbert problem
<math display="block">f(x) = F_{+}(x) - F_{-}(x)</math>

then the Hilbert transform of {{math|''f''(''x'')}} is given by{{sfn|Pandey|1996|loc=Chapter 2}}
<math display="block">H(f)(x) = -i \bigl( F_{+}(x) + F_{-}(x) \bigr) .</math>