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

$D_k(x)=\sum _{s=-k}^k{\mathrm{e}}^{isx}$

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 $F_n(x)\ge 0$ with average value of $1$. The convolution F_n is positive: for $f\ge 0$ of period $2\pi$ it satisfies

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

and, by Young's inequality,

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

for continuous function $f$; moreover,

$f\ast F_n\to f$ for every $f\in L^p([-\pi ,\pi ])$ ($1\le p<\mathrm{\infty }$)

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