Dirichlet kernel

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)