Dirichlet kernel

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

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

as required.

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:

See also


  • 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) [1994], "Dirichlet kernel", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  • Dirichlet-Kernel at PlanetMath
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