Fejér kernel

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.

The Fejér kernel is defined as

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

where $D_k\left(x\right)$ is the "k"th order Dirichlet kernel. It can also be written in a closed form as

:$F_n\left(x\right) = frac\left\{1\right\}\left\{n\right\} left\left(frac\left\{sin frac\left\{n x\right\}\left\{2\left\{sin frac\left\{x\right\}\left\{2 ight\right)^2$,

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

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

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

and, by the Hölder's inequality, $|F_n*f |_\left\{L^p\left( \left[-pi, pi\right] \right)\right\} le |f|_\left\{L^p\left( \left[-pi, pi\right] \right)\right\}$ for every $0 le p le infty$ or continuous function $f$;moreover, $f*F_n ightarrow f$ for every $f in L^p\left( \left[-pi, pi\right] \right)$ ($1 le p < infty$) or continuous function $f$.

ee also

* Fejér's theorem
* Gibbs phenomenon
* Charles Jean de la Vallée-Poussin

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