Μ operator

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

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

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

The important property of the Fejér kernel is $F_{n}(x)\geq 0$ with average value of $1$. The convolution F_n is positive: for $f\geq 0$ of period $2\pi$ it satisfies

$0\leq (f*F_{n})(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(y)F_{n}(x-y)\,dy,$

and, by Young's inequality,

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

for continuous function $f$; moreover,

$f*F_{n}\rightarrow f$ for every $f\in L^{p}([-\pi ,\pi ])$ ($1\leq p<\infty$)

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

See also

• Fejér's theorem
• Dirichlet kernel
• Gibbs phenomenon
• Charles Jean de la Vallée-Poussin