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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
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<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>
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{{TEX|auto}}
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{{TEX|done}}
  
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" />.
+
One of the integral transforms (cf. [[Integral transform|Integral transform]]) related to the [[Fourier transform|Fourier transform]]. Let the function $  F $
 +
have bounded variation on  $  (- \infty , + \infty ) $.  
 +
The function
  
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
+
$$ \tag{* }
 +
\phi ( x) = \
 +
{
 +
\frac{1}{\sqrt {2 \pi } }
 +
}
 +
\int\limits _ {- \infty } ^ {+\infty }
 +
e  ^ {-} ixy  dF ( y)
 +
$$
  
<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>
+
is called the Fourier–Stieltjes transform of  $  F $.
 +
The function  $  \phi $
 +
determined by the integral (*) is bounded and continuous. Every periodic function  $  \phi $
 +
that can be expanded in an absolutely-convergent Fourier series  $  \sum _ {- \infty }  ^ {+ \infty } a _ {n} e  ^ {inx} $
 +
can be written as an integral (*) with  $  F ( x) = \sum _ {n \leq  x }  a _ {n} $.
 +
 
 +
Formula (*) can be inverted: If  $  F $
 +
has bounded variation and if
 +
 
 +
$$
 +
F ^  \bullet  ( x)  = \
 +
{
 +
\frac{F ( x + 0) + F ( x - 0) }{2}
 +
} ,
 +
$$
  
 
then
 
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>
+
$$
 +
F ^  \bullet  ( x) - F ^  \bullet  ( 0= \
 +
{
 +
\frac{1}{\sqrt {2 \pi } }
 +
}
 +
\int\limits _ {- \infty } ^ {+\infty }
 +
\phi ( \xi )
 +
 
 +
\frac{e ^ {i \xi x } - 1 }{i \xi }
 +
\
 +
d \xi ,\ \
 +
x \in (- \infty , + \infty ),
 +
$$
  
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" />.
+
where the integral is taken to mean the principal value at $  \infty $.
  
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" />,
+
If one only allows non-decreasing functions of bounded variation as the function $  F $
 +
in formula (*), then the set of continuous functions $  \phi $
 +
thus obtained is completely characterized by the property: For any system of real numbers $  t _ {1} \dots t _ {n} $,
  
<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>
+
$$
 +
\sum _ {i, j = 1 } ^ { n }
 +
\phi ( t _ {i} - t _ {j} )
 +
\xi _ {i} \overline \xi \; _ {j}  \geq  0,
 +
$$
  
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
+
whatever the complex numbers $  \xi _ {1} \dots \xi _ {n} $(
 +
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>
+
$$
 +
P ( x)  = \
 +
{
 +
\frac{1}{\sqrt {2 \pi } }
 +
}
 +
F ( x)
 +
$$
  
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
+
is subjected to the additional restrictions $  \lim\limits _ {x \rightarrow - \infty }  P ( x) = 0 $,  
 +
$  \lim\limits _ {x \rightarrow + \infty }  P ( x) = 1 $
 +
and $  P $
 +
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>
+
$$
 +
\Phi(x) = \int\limits_{-\infty}^{+\infty} e^{ixy} dP(y)
 +
$$
  
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.
+
is called the characteristic function (of the distribution $  P  $).  
 +
The Bochner–Khinchin theorem then expresses a necessary and sufficient condition for a continuous function $  \Phi $(
 +
for which $  \Phi ( 0) = 1 $)  
 +
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.
+
The Fourier–Stieltjes transform has also been developed in the $  n $-
 +
dimensional case.
  
 
====References====
 
====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>
+
<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,  [[Gnedenko, "A course in the theory of probability"|"The theory of probability"]], Chelsea, reprint  (1962)  (Translated from Russian)</TD></TR>
 +
</table>

Latest revision as of 20:16, 16 January 2024


One of the integral transforms (cf. Integral transform) related to the Fourier transform. Let the function $ F $ have bounded variation on $ (- \infty , + \infty ) $. The function

$$ \tag{* } \phi ( x) = \ { \frac{1}{\sqrt {2 \pi } } } \int\limits _ {- \infty } ^ {+\infty } e ^ {-} ixy dF ( y) $$

is called the Fourier–Stieltjes transform of $ F $. The function $ \phi $ determined by the integral (*) is bounded and continuous. Every periodic function $ \phi $ that can be expanded in an absolutely-convergent Fourier series $ \sum _ {- \infty } ^ {+ \infty } a _ {n} e ^ {inx} $ can be written as an integral (*) with $ F ( x) = \sum _ {n \leq x } a _ {n} $.

Formula (*) can be inverted: If $ F $ has bounded variation and if

$$ F ^ \bullet ( x) = \ { \frac{F ( x + 0) + F ( x - 0) }{2} } , $$

then

$$ F ^ \bullet ( x) - F ^ \bullet ( 0) = \ { \frac{1}{\sqrt {2 \pi } } } \int\limits _ {- \infty } ^ {+\infty } \phi ( \xi ) \frac{e ^ {i \xi x } - 1 }{i \xi } \ d \xi ,\ \ x \in (- \infty , + \infty ), $$

where the integral is taken to mean the principal value at $ \infty $.

If one only allows non-decreasing functions of bounded variation as the function $ F $ in formula (*), then the set of continuous functions $ \phi $ thus obtained is completely characterized by the property: For any system of real numbers $ t _ {1} \dots t _ {n} $,

$$ \sum _ {i, j = 1 } ^ { n } \phi ( t _ {i} - t _ {j} ) \xi _ {i} \overline \xi \; _ {j} \geq 0, $$

whatever the complex numbers $ \xi _ {1} \dots \xi _ {n} $( 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

$$ P ( x) = \ { \frac{1}{\sqrt {2 \pi } } } F ( x) $$

is subjected to the additional restrictions $ \lim\limits _ {x \rightarrow - \infty } P ( x) = 0 $, $ \lim\limits _ {x \rightarrow + \infty } P ( x) = 1 $ and $ P $ is continuous on the left; it is called a distribution, and

$$ \Phi(x) = \int\limits_{-\infty}^{+\infty} e^{ixy} dP(y) $$

is called the characteristic function (of the distribution $ P $). The Bochner–Khinchin theorem then expresses a necessary and sufficient condition for a continuous function $ \Phi $( for which $ \Phi ( 0) = 1 $) to be the characteristic function of a certain distribution.

The Fourier–Stieltjes transform has also been developed in the $ n $- dimensional case.

References

[1] S. Bochner, "Lectures on Fourier integrals" , Princeton Univ. Press (1959) (Translated from German)
[2] A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)
[3] B.V. Gnedenko, "The theory of probability", Chelsea, reprint (1962) (Translated from Russian)
How to Cite This Entry:
Fourier-Stieltjes transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier-Stieltjes_transform&oldid=16936
This article was adapted from an original article by P.I. Lizorkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article