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Difference between revisions of "Dirichlet kernel"

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====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>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  P.G.L. Dirichlet, "Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données"  ''J. für Math.'' , '''4'''  (1829)  pp. 157–169</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  P.G.L. Dirichlet, "Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données" , ''Werke'' , '''1''' , Chelsea, reprint  (1969)  pp. 117–132</TD></TR
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><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>
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<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>
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<TR><TD valign="top">[5]</TD> <TD valign="top">  A. Zygmund, "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR>
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</table>
  
 
====Comments====
 
====Comments====
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$$  
 
$$  
\sum _ { k=- } n ^ { n }  e  ^ {ikx} \ \  
+
\sum _ { k=-n }^{ n }  e  ^ {ikx} \ \  
 
\textrm{ and } \ \  
 
\textrm{ and } \ \  
\sum _ { k=- } n ^ { n }   
+
\sum _ { k=-n }^{ n }   
 
\frac{ \mathop{\rm sgn}  n }{i}
 
\frac{ \mathop{\rm sgn}  n }{i}
 
  e  ^ {ikx} ,
 
  e  ^ {ikx} ,

Latest revision as of 09:17, 9 June 2024


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 trigonométriques qui servent à représenter 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 trigonométriques qui servent à représenter 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=46719
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article