Dirichlet kernel

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In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as

where n is any nonnegative integer. The kernel functions are periodic with period .

Plot restricted to one period of the first few Dirichlet kernels showing their convergence to one of the Dirac delta distributions of the Dirac comb.

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

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.

Plot restricted to one period of the first few Dirichlet kernels (multiplied by ).

L1 norm of the kernel function[edit]

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:

where is the sine integral

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 periodic delta function[edit]

The Dirichlet kernel is a periodic function which becomes the Dirac comb, i.e. the periodic delta function, in the limit

with the angular frequency .

This can be inferred from the autoconjugation property of the Dirichlet kernel under forward and inverse Fourier transform:

and goes to the Dirac comb of period as , which remains invariant under Fourier transform: . Thus must also have converged to as .

In a different vein, consider ∆(x) as 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 in pointwise convergence mentioned above).

Proof of the trigonometric identity[edit]

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[edit]

Start with the series

Multiply both sides by and use the trigonometric identity
to reduce the terms in the sum.
which telescopes down to the result.

Variant of identity[edit]

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:

Another variant is and this can be easily proved by using an identity .[1]

See also[edit]


  1. ^ Fay, Temple H.; Kloppers, P. Hendrik (2001). "The Gibbs' phenomenon". International Journal of Mathematical Education in Science and Technology. 32 (1): 73–89. doi:10.1080/00207390117151. S2CID 120595055.