# Fejér kernel

In mathematics, the Fejér kernel is used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity.

Plot of several Fejér kernels

The Fejér kernel is defined as

$F_n(x) = \frac{1}{n} \sum_{k=0}^{n-1}D_k(x),$

where

$D_k(x)=\sum_{s=-k}^k {\rm e}^{isx}$

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

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

where this expression is defined.[1] It is named after the Hungarian mathematician Lipót Fejér (1880–1959).

The important property of the Fejér kernel is $F_n(x) \ge 0$. The convolution Fn is positive: for $f \ge 0$ of period $2 \pi$ it satisfies

$0 \le (f*F_n)(x)=\frac{1}{2\pi}\int_{-\pi}^\pi f(y) F_n(x-y)\,dy,$

and, by Young's inequality,

$\|F_n*f \|_{L^p([-\pi, \pi])} \le \|f\|_{L^p([-\pi, \pi])}$ for every $0 \le p \le \infty$

for continuous function $f$; moreover,

$f*F_n \rightarrow f$ for every $f \in L^p([-\pi, \pi])$ ($1 \le p < \infty$)

for continuous function $f$. Indeed, if $f$ is continuous, then the convergence is uniform.