Editing Hilbert transform (section)

=== 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>