where n is any nonnegative integer. The kernel functions are periodic with period .
Plot of the first few Dirichlet kernels showing its convergence to the Dirac delta distribution.
The importance of the Dirichlet kernel comes from its relation to Fourier series. The convolution of Dn(x) with any function f of period 2π is the nth-degree Fourier series approximation to f, i.e., we have
where
is the kth Fourier coefficient of f. This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel.
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 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.
Take the periodicDirac delta function defined by ∆(x) = 𝛿(floor(x/pi)), which is not a function of a real variable, but rather a "generalized function", also called a "distribution", and multiply by 2π. (Here 𝛿(x) denotes the Dirichlet delta function.) We get the identity element for convolution on functions of period 2π. In other words, we have
for every function f of period 2π. The Fourier series representation of this "function" is
(This Fourier series converges to the function almost nowhere.) 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).
If the sum is only over non negative integers (which may arise when computing a discrete Fourier transform that is not centered), then using similar techniques we can show the following identity:
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