# Fejér kernel

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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

## Definition

The Fejér kernel is defined as

${\displaystyle F_{n}(x)={\frac {1}{n}}\sum _{k=0}^{n-1}D_{k}(x),}$

where

${\displaystyle D_{k}(x)=\sum _{s=-k}^{k}{\rm {e}}^{isx}}$

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

${\displaystyle F_{n}(x)={\frac {1}{n}}\left({\frac {\sin {\frac {nx}{2}}}{\sin {\frac {x}{2}}}}\right)^{2}={\frac {1}{n}}\left({\frac {1-\cos(nx)}{1-\cos x}}\right)}$,

where this expression is defined.[1]

The Fejér kernel can also be expressed as

${\displaystyle F_{n}(x)=\sum _{|j|\leq n}\left(1-{\frac {|j|}{n}}\right)e^{ijx}}$.

## Properties

The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is ${\displaystyle F_{n}(x)\geq 0}$ with average value of ${\displaystyle 1}$.

### Convolution

The convolution Fn is positive: for ${\displaystyle f\geq 0}$ of period ${\displaystyle 2\pi }$ it satisfies

${\displaystyle 0\leq (f*F_{n})(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(y)F_{n}(x-y)\,dy.}$
${\displaystyle \|F_{n}*f\|_{L^{p}([-\pi ,\pi ])}\leq \|f\|_{L^{p}([-\pi ,\pi ])}}$ for every ${\displaystyle 1\leq p\leq \infty }$

Additionally, if ${\displaystyle f\in L^{1}([-\pi ,\pi ])}$, then

${\displaystyle f*F_{n}\rightarrow f}$ a.e.

If ${\displaystyle f}$ is continuous, then the convergence is uniform, yielding a proof of the Weierstrass theorem.

## References

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