Dirichlet kernel
The expression
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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:
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The integral on the right-hand side is said to be Dirichlet's singular integral.
In analogy with the Dirichlet kernel [3], the expression
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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:
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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
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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) |
Dirichlet kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_kernel&oldid=19301