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A series
 
A series
  
<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/f041130/f0411301.png" /></td> </tr></table>
+
$$
 +
{
 +
\frac{a _ {0} }{2}
 +
} +
 +
\sum _ {n = 1 } ^  \infty 
 +
( a _ {n}  \cos  nx + b _ {n}  \sin  nx),
 +
$$
  
where for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f0411302.png" />
+
where for $  n = 0, 1 \dots $
  
<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/f041130/f0411303.png" /></td> </tr></table>
+
$$
 +
a _ {n}  = \
 +
{
 +
\frac{1} \pi
 +
}
 +
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
\cos  nx  dF ( x),\ \
 +
b _ {n}  = \
 +
{
 +
\frac{1} \pi
 +
}
 +
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
\sin  nx  dF ( x)
 +
$$
  
(the integrals are taken in the sense of Stieltjes). Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f0411304.png" /> is a function of bounded variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f0411305.png" />. Alternatively one could write
+
(the integrals are taken in the sense of Stieltjes). Here $  F $
 +
is a function of bounded variation on $  [ 0, 2 \pi ] $.  
 +
Alternatively one could write
  
<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/f041130/f0411306.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
dF ( x)  \sim \
 +
{
 +
\frac{a _ {0} }{2}
 +
} +
 +
\sum _ {n = 1 } ^  \infty 
 +
( a _ {n}  \cos  nx + b _ {n}  \sin  nx).
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f0411307.png" /> is absolutely continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f0411308.png" />, then (*) is the Fourier series of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f0411309.png" />. In complex form the series (*) is
+
If $  F $
 +
is absolutely continuous on $  [ 0, 2 \pi ] $,
 +
then (*) is the Fourier series of the function $  F ^ { \prime } $.  
 +
In complex form the series (*) is
  
<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/f041130/f04113010.png" /></td> </tr></table>
+
$$
 +
dF ( x)  \sim \
 +
\sum _ {n = - \infty } ^ { {+ }  \infty }
 +
c _ {n} e  ^ {inx} ,
 +
$$
  
 
where
 
where
  
<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/f041130/f04113011.png" /></td> </tr></table>
+
$$
 +
c _ {n}  = \
 +
{
 +
\frac{1}{2 \pi }
 +
}
 +
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
e  ^ {-} inx  dF ( x).
 +
$$
  
 
Moreover,
 
Moreover,
  
<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/f041130/f04113012.png" /></td> </tr></table>
+
$$
 +
F ( x) - c _ {0} x  \sim \
 +
C _ {0} + \sum _ {
 +
\begin{array}{c}
 +
n = - \infty \\
 +
n \neq 0  
 +
\end{array}
 +
} ^  \infty 
 +
 
 +
\frac{c _ {n} }{in }
 +
 
 +
e  ^ {inx} ,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f04113013.png" /> will be bounded. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f04113014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f04113015.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f04113016.png" />. There is a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f04113017.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f04113018.png" /> does not tend to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f04113019.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f04113020.png" />. The series (*) is summable to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f04113021.png" /> by the Cesàro method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f04113022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f04113023.png" />, almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041130/f04113024.png" />.
+
and $  \{ c _ {n} \} $
 +
will be bounded. If $  c _ {n} \rightarrow 0 $,  
 +
then $  F $
 +
is continuous on $  [ 0, 2 \pi ] $.  
 +
There is a continuous function $  F $
 +
for which $  c _ {n} $
 +
does not tend to 0 $
 +
as $  n \rightarrow + \infty $.  
 +
The series (*) is summable to $  F ^ { \prime } ( x) $
 +
by the Cesàro method $  ( C, r) $,
 +
$  r > 0 $,  
 +
almost-everywhere on $  [ 0, 2 \pi ] $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR></table>

Latest revision as of 19:39, 5 June 2020


A series

$$ { \frac{a _ {0} }{2} } + \sum _ {n = 1 } ^ \infty ( a _ {n} \cos nx + b _ {n} \sin nx), $$

where for $ n = 0, 1 \dots $

$$ a _ {n} = \ { \frac{1} \pi } \int\limits _ { 0 } ^ { {2 } \pi } \cos nx dF ( x),\ \ b _ {n} = \ { \frac{1} \pi } \int\limits _ { 0 } ^ { {2 } \pi } \sin nx dF ( x) $$

(the integrals are taken in the sense of Stieltjes). Here $ F $ is a function of bounded variation on $ [ 0, 2 \pi ] $. Alternatively one could write

$$ \tag{* } dF ( x) \sim \ { \frac{a _ {0} }{2} } + \sum _ {n = 1 } ^ \infty ( a _ {n} \cos nx + b _ {n} \sin nx). $$

If $ F $ is absolutely continuous on $ [ 0, 2 \pi ] $, then (*) is the Fourier series of the function $ F ^ { \prime } $. In complex form the series (*) is

$$ dF ( x) \sim \ \sum _ {n = - \infty } ^ { {+ } \infty } c _ {n} e ^ {inx} , $$

where

$$ c _ {n} = \ { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } e ^ {-} inx dF ( x). $$

Moreover,

$$ F ( x) - c _ {0} x \sim \ C _ {0} + \sum _ { \begin{array}{c} n = - \infty \\ n \neq 0 \end{array} } ^ \infty \frac{c _ {n} }{in } e ^ {inx} , $$

and $ \{ c _ {n} \} $ will be bounded. If $ c _ {n} \rightarrow 0 $, then $ F $ is continuous on $ [ 0, 2 \pi ] $. There is a continuous function $ F $ for which $ c _ {n} $ does not tend to $ 0 $ as $ n \rightarrow + \infty $. The series (*) is summable to $ F ^ { \prime } ( x) $ by the Cesàro method $ ( C, r) $, $ r > 0 $, almost-everywhere on $ [ 0, 2 \pi ] $.

References

[1] A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)
How to Cite This Entry:
Fourier-Stieltjes series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier-Stieltjes_series&oldid=22447
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article