Namespaces
Variants
Actions

Difference between revisions of "Dirichlet kernel"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
d0328801.png
 +
$#A+1 = 13 n = 0
 +
$#C+1 = 13 : ~/encyclopedia/old_files/data/D032/D.0302880 Dirichlet kernel
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
The expression
 
The expression
  
<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/d/d032/d032880/d0328801.png" /></td> </tr></table>
+
$$
 +
D _ {n} ( x)  =
 +
\frac{1}{2}
 +
+ \sum _ {k = 1 } ^ { n }
 +
\cos  k x  =
 +
\frac{\sin ( n + 1 / 2 ) x }{2 \sin  x / 2 }
 +
.
 +
$$
 +
 
 +
It was shown by P.G.L. Dirichlet [[#References|[1]]] that the partial sum  $  S _ {n} $
 +
of the Fourier series of a function  $  f $
 +
is expressed by the Dirichlet kernel:
  
It was shown by P.G.L. Dirichlet [[#References|[1]]] that the partial sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032880/d0328802.png" /> of the Fourier series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032880/d0328803.png" /> is expressed by the Dirichlet kernel:
+
$$
 +
S _ {n} ( x)  =
 +
\frac{a _ {0} }{2}
 +
+ \sum _ {k = 1 } ^ { n }
 +
a _ {k}  \cos  k x + b _ {k}  \sin  k x =
 +
$$
  
<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/d/d032/d032880/d0328804.png" /></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/d/d032/d032880/d0328805.png" /></td> </tr></table>
+
\frac{1} \pi
 +
\int\limits _ {- \pi } ^  \pi  f ( t) D _ {n} ( t - x )  d t .
 +
$$
  
 
The integral on the right-hand side is said to be Dirichlet's singular integral.
 
The integral on the right-hand side is said to be Dirichlet's singular integral.
Line 13: Line 44:
 
In analogy with the Dirichlet kernel [[#References|[3]]], the expression
 
In analogy with the Dirichlet kernel [[#References|[3]]], the expression
  
<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/d/d032/d032880/d0328806.png" /></td> </tr></table>
+
$$
 +
\widetilde{D}  _ {n} ( x)  = \sum _ {k = 1 } ^ { n }
 +
\sin  k x  =
 +
\frac{\cos  x / 2 - \cos ( n + 1
 +
/ 2 ) x }{2 \sin  x / 2 }
 +
 
 +
$$
  
is called the conjugate Dirichlet kernel. The partial sum of the series conjugate with the Fourier series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032880/d0328807.png" /> is expressed by the conjugate Dirichlet kernel:
+
is called the conjugate Dirichlet kernel. The partial sum of the series conjugate with the Fourier series of a function $  f $
 +
is expressed by the conjugate Dirichlet kernel:
  
<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/d/d032/d032880/d0328808.png" /></td> </tr></table>
+
$$
 +
\widetilde{S}  _ {n} ( x)  = \sum _ {k = 1 } ^ { n }  b _ {k}  \cos  k x -
 +
a _ {k}  \sin  k x  = -
 +
\frac{1} \pi
 +
\int\limits _ {- \pi } ^  \pi 
 +
f ( t) \widetilde{D}  _ {n} ( t - x )  d t .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.G.L. Dirichlet,  "Sur la convergence des séries trigonometriques qui servent à réprésenter une fonction arbitraire entre des limites données"  ''J. für Math.'' , '''4'''  (1829)  pp. 157–169</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.G.L. Dirichlet,  "Sur la convergence des séries trigonometriques qui servent à réprésenter une fonction arbitraire entre des limites données" , ''Werke'' , '''1''' , Chelsea, reprint  (1969)  pp. 117–132</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Tauber,  "Ueber den Zusammenhang des reellen und imaginären Teiles einer Potentzreihe"  ''Monatsh. Math.'' , '''2'''  (1891)  pp. 79–118</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.G.L. Dirichlet,  "Sur la convergence des séries trigonometriques qui servent à réprésenter une fonction arbitraire entre des limites données"  ''J. für Math.'' , '''4'''  (1829)  pp. 157–169</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.G.L. Dirichlet,  "Sur la convergence des séries trigonometriques qui servent à réprésenter une fonction arbitraire entre des limites données" , ''Werke'' , '''1''' , Chelsea, reprint  (1969)  pp. 117–132</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Tauber,  "Ueber den Zusammenhang des reellen und imaginären Teiles einer Potentzreihe"  ''Monatsh. Math.'' , '''2'''  (1891)  pp. 79–118</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The Dirichlet kernel is also called the Dirichlet summation kernel. There is also a different normalization in use: the kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032880/d0328809.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032880/d03288010.png" /> are often multiplied by 2. They are then represented also by the series
+
The Dirichlet kernel is also called the Dirichlet summation kernel. There is also a different normalization in use: the kernels $  D _ {n} $
 +
and $  \widetilde{D}  _ {n} $
 +
are often multiplied by 2. They are then represented also by the 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/d/d032/d032880/d03288011.png" /></td> </tr></table>
+
$$
 +
\sum _ { k=- } n ^ { n }  e  ^ {ikx} \ \
 +
\textrm{ and } \ \
 +
\sum _ { k=- } n ^ { n } 
 +
\frac{ \mathop{\rm sgn}  n }{i}
 +
e  ^ {ikx} ,
 +
$$
  
respectively. The factors preceding the two integrals in the main article above then become <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032880/d03288012.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032880/d03288013.png" />.
+
respectively. The factors preceding the two integrals in the main article above then become $  \pm  1 / 2 \pi $
 +
instead of $  \pm  1 / \pi $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Dym,  H.P. McKean,  "Fourier series and integrals" , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Dym,  H.P. McKean,  "Fourier series and integrals" , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR></table>

Revision as of 19:35, 5 June 2020


The expression

$$ D _ {n} ( x) = \frac{1}{2} + \sum _ {k = 1 } ^ { n } \cos k x = \frac{\sin ( n + 1 / 2 ) x }{2 \sin x / 2 } . $$

It was shown by P.G.L. Dirichlet [1] that the partial sum $ S _ {n} $ of the Fourier series of a function $ f $ is expressed by the Dirichlet kernel:

$$ S _ {n} ( x) = \frac{a _ {0} }{2} + \sum _ {k = 1 } ^ { n } a _ {k} \cos k x + b _ {k} \sin k x = $$

$$ = \ \frac{1} \pi \int\limits _ {- \pi } ^ \pi f ( t) D _ {n} ( t - x ) d t . $$

The integral on the right-hand side is said to be Dirichlet's singular integral.

In analogy with the Dirichlet kernel [3], the expression

$$ \widetilde{D} _ {n} ( x) = \sum _ {k = 1 } ^ { n } \sin k x = \frac{\cos x / 2 - \cos ( n + 1 / 2 ) x }{2 \sin x / 2 } $$

is called the conjugate Dirichlet kernel. The partial sum of the series conjugate with the Fourier series of a function $ f $ is expressed by the conjugate Dirichlet kernel:

$$ \widetilde{S} _ {n} ( x) = \sum _ {k = 1 } ^ { n } b _ {k} \cos k x - a _ {k} \sin k x = - \frac{1} \pi \int\limits _ {- \pi } ^ \pi f ( t) \widetilde{D} _ {n} ( t - x ) d t . $$

References

[1] P.G.L. Dirichlet, "Sur la convergence des séries trigonometriques qui servent à réprésenter une fonction arbitraire entre des limites données" J. für Math. , 4 (1829) pp. 157–169
[2] P.G.L. Dirichlet, "Sur la convergence des séries trigonometriques qui servent à réprésenter une fonction arbitraire entre des limites données" , Werke , 1 , Chelsea, reprint (1969) pp. 117–132
[3] A. Tauber, "Ueber den Zusammenhang des reellen und imaginären Teiles einer Potentzreihe" Monatsh. Math. , 2 (1891) pp. 79–118
[4] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[5] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)

Comments

The Dirichlet kernel is also called the Dirichlet summation kernel. There is also a different normalization in use: the kernels $ D _ {n} $ and $ \widetilde{D} _ {n} $ are often multiplied by 2. They are then represented also by the series

$$ \sum _ { k=- } n ^ { n } e ^ {ikx} \ \ \textrm{ and } \ \ \sum _ { k=- } n ^ { n } \frac{ \mathop{\rm sgn} n }{i} e ^ {ikx} , $$

respectively. The factors preceding the two integrals in the main article above then become $ \pm 1 / 2 \pi $ instead of $ \pm 1 / \pi $.

References

[a1] H. Dym, H.P. McKean, "Fourier series and integrals" , Acad. Press (1972)
[a2] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)
How to Cite This Entry:
Dirichlet kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_kernel&oldid=19301
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article