Fejér kernel

In mathematics, a periodic continuous function, that is, the -th Fejér polynomial defined by

In which

The -th Fourier coefficient. With the help of these trigonometric polynomials Fejér gave a constructive proof of the theorem of Weierstrass, which says that any periodic continuous function can be uniformly approximated by trigonometric polynomials. This statement is also referred to as a set of Fejér.

Convergence statements - Set of Fejér

Fejér led the proof of the (first) arithmetic mean of the partial sums of the Fourier series

In which

The - th partial sum is by showing:

Converges for each periodic continuous function, the result of Fejér polynomials uniformly against, ie

Fejér kernel

The nth Fejer core is defined by

Convolution

The Fejér polynomials can be represented as a convolution with the Fejér kernel. It is

Arithmetic mean of the Dirichlet kernel

From the interpretation of the Fejér polynomials as (first ) the arithmetic mean of the partial sums follows the presentation of the Fejér kernel as the arithmetic mean of the Dirichlet kernel

Where the Dirichlet kernel is defined by

Positive real core

In addition to the sum notation of complex functions, the Fejér kernel can be represented in a closed form. For this purpose is used that the Dirichlet kernel representation

Possesses. Using the above relationship of Fejér kernel with Dirichlet kernels and usually

Results in the following closed-form representation of the Fejér kernel.

Given the apparent positivity of the Fejér kernel of Fejér polynomials, the set of Bohman - Korowkin can be applied for the detection of uniform convergence, which states that it follows from the uniform convergence of the test functions and the uniform convergence for all functions.

Convergence in other function spaces

Also for non-continuous functions of other function rooms, for example, the Lebesgue - integrable functions, statements to be specify to approximability.

Quantitative statements

For Hölder continuous functions to direct estimates of the convergence behavior of Fejér polynomials can be stated.

Belongs to the class of a Hölder - continuous functions, ie

So the following quantitative Approximationsaussagen apply: