Dirichlet kernel

The Dirichlet kernel is an examined by Peter Gustav Lejeune Dirichlet function sequence. This is used in the analysis in the branch of Fourier analysis. Dirichlet was in 1829 the first rigorous proof of the convergence of the Fourier series of a periodic, piecewise continuous and piecewise monotone function. The convergence of Fourier series has been discussed since Leonhard Euler. This sequence of functions found by Dirichlet is an important part of this proof, where it is used as an integral kernel. That's why they are called Dirichlet kernel.

Definition

As the Dirichlet kernel is defined as the sequence of functions

The importance of the Dirichlet kernel is related to the ratio of the Fourier series. The folding of Dn (x) with a function f of the period 2π is the n-th degree of its Fourier series approximation for f example,

In which

Of the k-th Fourier coefficient of F. This suggests that it is sufficient to study the convergence of Fourier series, to study the properties of the Dirichlet kernel. From the fact that the L1 norm of Dn is compared to a logarithmic one can deduce that there is a continuous function that is not represented by their Fourier series. Explicitly applies namely:

For the notation see Landau symbols.

Relationship to the delta distribution

The periodic delta function is the identity element for the convolution with - periodic functions:

F for each function with period. The Fourier series is represented by the following " function":

Proof of the trigonometric identity

The trigonometric identity

Can be demonstrated as follows. These Visualise you look at the finite sum of the geometric series:

In particular

Multiplying numerator and denominator by r-1 /2, one obtains

In the case of r = eix obtained

And eventually shortens by " -2i ".