In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipót Fejér (1880–1959).

Plot of several Fejér kernels

Definition

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The Fejér kernel has many equivalent definitions. We outline three such definitions below:

1) The traditional definition expresses the Fejér kernel   in terms of the Dirichlet kernel:  

where

 

is the kth order Dirichlet kernel.

2) The Fejér kernel   may also be written in a closed form expression as follows[1]

 

This closed form expression may be derived from the definitions used above. The proof of this result goes as follows.

First, we use the fact that the Dirichlet kernel may be written as:[2]

 

Hence, using the definition of the Fejér kernel above we get:

 

Using the trigonometric identity:  

 

Hence it follows that:

 

3) The Fejér kernel can also be expressed as:

 

Properties

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The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is   with average value of  .

Convolution

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The convolution Fn is positive: for   of period   it satisfies

 

Since  , we have  , which is Cesàro summation of Fourier series.

By Young's convolution inequality,

 

Additionally, if  , then

  a.e.

Since   is finite,  , so the result holds for other   spaces,   as well.

If   is continuous, then the convergence is uniform, yielding a proof of the Weierstrass theorem.

  • One consequence of the pointwise a.e. convergence is the uniqueness of Fourier coefficients: If   with  , then   a.e. This follows from writing  , which depends only on the Fourier coefficients.
  • A second consequence is that if   exists a.e., then   a.e., since Cesàro means   converge to the original sequence limit if it exists.

Applications

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The Fejér kernel is used in signal processing and Fourier analysis.

See also

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References

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  1. ^ Hoffman, Kenneth (1988). Banach Spaces of Analytic Functions. Dover. p. 17. ISBN 0-486-45874-1.
  2. ^ Konigsberger, Konrad. Analysis 1 (in German) (6th ed.). Springer. p. 322.