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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 [[#References|[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 [[#References|[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====
<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====
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====
<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>