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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

Contents

DefinitionEdit

The Fejér kernel is defined as

 

where

 

is the kth order Dirichlet kernel. It can also be written in a closed form as

 ,

where this expression is defined.[1]

The Fejér kernel can also be expressed as

 .

PropertiesEdit

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

ConvolutionEdit

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,

  for every  

for  .

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.

See alsoEdit

ReferencesEdit

  1. ^ Hoffman, Kenneth (1988). Banach Spaces of Analytic Functions. Dover. p. 17. ISBN 0-486-45874-1.