In mathematical analysis, the Dirichlet kernel is the collection of functions
It is named after Peter Gustav Lejeune Dirichlet.
The importance of the Dirichlet kernel comes from its relation to Fourier series. The convolution of Dn(x) with any function ƒ of period 2π is the nth-degree Fourier series approximation to ƒ, i.e., we have
is the kth Fourier coefficient of ƒ. This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel.
L1 norm of the kernel function
Of particular importance is the fact that the L1 norm of Dn on diverges to infinity as n → ∞. One can estimate that
By using a Riemann-sum argument to estimate the contribution in the largest neighbourhood of zero in which is positive, and the Jensen's inequality for the remaining part, it is also possible to show that:
This lack of uniform integrability is behind many divergence phenomena for the Fourier series. For example, together with the uniform boundedness principle, it can be used to show that the Fourier series of a continuous function may fail to converge pointwise, in rather dramatic fashion. See convergence of Fourier series for further details.
A precise proof of the first result that is given by
where we have used the Taylor series identity that and where are the first-order harmonic numbers.
Relation to the delta function
Take the periodic Dirac delta function, which is not a function of a real variable, but rather a "generalized function", also called a "distribution", and multiply by 2π. We get the identity element for convolution on functions of period 2π. In other words, we have
for every function ƒ of period 2π. The Fourier series representation of this "function" is
Therefore the Dirichlet kernel, which is just the sequence of partial sums of this series, can be thought of as an approximate identity. Abstractly speaking it is not however an approximate identity of positive elements (hence the failures mentioned above).
Proof of the trigonometric identity
displayed at the top of this article may be established as follows. First recall that the sum of a finite geometric series is
In particular, we have
Multiply both the numerator and the denominator by , getting
In the case we have
Alternative proof of the trigonometric identity
Start with the series
Multiply both sides of the above by
and use the trigonometric identity
to reduce the right-hand side to
Variant of identity
If the sum is only over non negative integers (which may arise when computing a DFT that is not centered), then using similar techniques we can show the following identity:
- Andrew M. Bruckner, Judith B. Bruckner, Brian S. Thomson: Real Analysis. ClassicalRealAnalysis.com 1996, ISBN 0-13-458886-X, S.620 (vollständige Online-Version (Google Books))
- Podkorytov, A. N. (1988), "Asymptotic behavior of the Dirichlet kernel of Fourier sums with respect to a polygon". Journal of Soviet Mathematics, 42(2): 1640–1646. doi: 10.1007/BF01665052
- Levi, H. (1974), "A geometric construction of the Dirichlet kernel". Transactions of the New York Academy of Sciences, 36: 640–643. doi: 10.1111/j.2164-0947.1974.tb03023.x
- Hazewinkel, Michiel, ed. (2001) , "Dirichlet kernel", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Dirichlet-Kernel at PlanetMath