Dirichlet kernel

The expression

It was shown by P.G.L. Dirichlet [1] that the partial sum of the Fourier series of a function is expressed by the Dirichlet kernel:

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

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

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

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 and are often multiplied by 2. They are then represented also by the series

respectively. The factors preceding the two integrals in the main article above then become instead of .

References

[a1]	H. Dym, H.P. McKean, "Fourier series and integrals" , Acad. Press (1972)
[a2]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)