View source for Fourier-Stieltjes transform

One of the integral transforms (cf. [[Integral transform|Integral transform]]) related to the [[Fourier transform|Fourier transform]]. Let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f0411401.png" /> have bounded variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f0411402.png" />. The function

<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/f/f041/f041140/f0411403.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

is called the Fourier–Stieltjes transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f0411404.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f0411405.png" /> determined by the integral (*) is bounded and continuous. Every periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f0411406.png" /> that can be expanded in an absolutely-convergent Fourier series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f0411407.png" /> can be written as an integral (*) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f0411408.png" />.

Formula (*) can be inverted: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f0411409.png" /> has bounded variation and if

<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/f/f041/f041140/f04114010.png" /></td> </tr></table>

then

<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/f/f041/f041140/f04114011.png" /></td> </tr></table>

where the integral is taken to mean the principal value at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f04114012.png" />.

If one only allows non-decreasing functions of bounded variation as the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f04114013.png" /> in formula (*), then the set of continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f04114014.png" /> thus obtained is completely characterized by the property: For any system of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f04114015.png" />,

<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/f/f041/f041140/f04114016.png" /></td> </tr></table>

whatever the complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f04114017.png" /> (the Bochner–Khinchin theorem). Such functions are called positive definite. The Fourier–Stieltjes transform is extensively applied in probability theory, where the non-decreasing function

<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/f/f041/f041140/f04114018.png" /></td> </tr></table>

is subjected to the additional restrictions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f04114019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f04114020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f04114021.png" /> is continuous on the left; it is called a distribution, 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/f/f041/f041140/f04114022.png" /></td> </tr></table>

is called the characteristic function (of the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f04114023.png" />). The Bochner–Khinchin theorem then expresses a necessary and sufficient condition for a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f04114024.png" /> (for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f04114025.png" />) to be the characteristic function of a certain distribution.

The Fourier–Stieltjes transform has also been developed in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041140/f04114026.png" />-dimensional case.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Bochner,   "Lectures on Fourier integrals" , Princeton Univ. Press  (1959)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.V. Gnedenko,   "The theory of probability" , Chelsea, reprint  (1962)  (Translated from Russian)</TD></TR></table>