View source for Hilbert transform

''of a function $f$''

The improper integral


\begin{equation}\label{eq:1}
g\left(x\right) = \frac{1}{\pi} \int_{0}^{\infty}\frac{f\left(x+t\right)-f\left(x-t\right)}{t}\mathrm{d}t.
\end{equation}

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h0474303.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h0474304.png" /> exists for almost-all values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h0474305.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h0474306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h0474307.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h0474308.png" /> also belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h0474309.png" /> and the inversion formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

is valid almost-everywhere. Here

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

where the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743012.png" /> depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743013.png" />.

Formulas (1) and (2) are equivalent to the formulas

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

in which the integrals are understood in the sense of the principal value.

The integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

understood in the sense of its principal value, is also called the Hilbert transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743017.png" />. This integral is often called the [[Hilbert singular integral|Hilbert singular integral]]. In the theory of Fourier series the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743018.png" /> defined by (6) is said to be conjugate with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743019.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743021.png" /> exists almost-everywhere, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743022.png" /> satisfies a Lipschitz condition of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743024.png" /> exists for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743025.png" /> and satisfies the same condition. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743027.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743028.png" /> has the same property, and an inequality analogous to (3) in which the integrals are taken over the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743029.png" /> is valid. Thus, the integral operators generated by the Hilbert transform are bounded (linear) operators on the respective spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743030.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743031.png" /> satisfies a Lipschitz condition, or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743032.png" />, and also

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743033.png" /></td> </tr></table>

the following inversion formula is valid:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743035.png" /></td> </tr></table>

In the class of functions which satisfy a Lipschitz condition, equation (7) is valid everywhere, and in the class of functions with integrable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743036.png" />-th power, it is valid almost-everywhere.

Each one of the pairs of formulas written above, such as (4) or (5), may be considered as an integral equation of the first kind, and the second formula yields the solution of this equation.

If the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743038.png" /> are considered as kernels of integral operators, they are often referred to as the [[Hilbert kernel|Hilbert kernel]] and as the [[Cauchy kernel|Cauchy kernel]]. In the case of the unit circle, there exists a simple relationship between these kernels:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743039.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047430/h04743041.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Hilbert,   "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen" , Chelsea, reprint  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Riesz,   "Sur les fonctions conjugées"  ''Math. Z.'' , '''27'''  (1927)  pp. 218–244</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.C. Titchmarsh,   "Introduction to the theory of Fourier integrals" , Oxford Univ. Press  (1948)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.I. Muskhelishvili,   "Singular integral equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR></table>



====Comments====
The defining integral could be represented in other ways. For further details one might refer to [http://en.wikipedia.org/wiki/Hilbert_transform Wikipedia] .

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>