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

where D_k(x) is the "k"th 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 ight)^2,

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. The convolution "Fn" is positive: for f ge 0 of period 2 pi it satisfies

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

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